Solution. This diagram can be expanded for functions of more than one variable, as we shall see very shortly. If $z=x y+f\left(x^2+y^2\right),$ show that y\frac{\partial z}{\partial x}-x\frac{\partial z}{\partial y}=y^2-x^2. Calculate $$\displaystyle ∂z/∂x,∂z/dy,dx/dt,$$ and $$\displaystyle dy/dt$$, then use Equation \ref{chain1}. We wish to prove that $$\displaystyle z=f(x(t),y(t))$$ is differentiable at $$\displaystyle t=t_0$$ and that Equation \ref{chain1} holds at that point as well. It is especially transparent using o() (Chain Rule Involving Two Independent Variables) Suppose z=f(x,y) is a differentiable function at (x,y) and that the partial derivatives of x=x(u,v) and y=y(u,v) exist at (u,v). Then the composite function z=f(x(u,v),y(u,v)) is differentiable at (u,v) with \begin{equation} \frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u} \qquad \text{and} \qquad \frac{\partial z}{\partial v}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial v}. To find the equation of the tangent line, we use the point-slope form (Figure $$\PageIndex{5}$$): \begin{align*} y−y_0 =m(x−x_0)\\[4pt]y−1 =\dfrac{7}{4}(x−2) \\[4pt] y =\dfrac{7}{4}x−\dfrac{7}{2}+1\\[4pt] y =\dfrac{7}{4}x−\dfrac{5}{2}.\end{align*}. Example 12.5.3 Using the Multivariable Chain Rule. \end{align*}\], The formulas for $$\displaystyle ∂w/∂u$$ and $$\displaystyle ∂w/∂v$$ are, \begin{align*} \dfrac{∂w}{∂u} =\dfrac{∂w}{∂x}⋅\dfrac{∂x}{∂u}+\dfrac{∂w}{∂y}⋅\dfrac{∂y}{∂u}+\dfrac{∂w}{∂z}⋅\dfrac{∂z}{∂u} \\[4pt] \dfrac{∂w}{∂v} =\dfrac{∂w}{∂x}⋅\dfrac{∂x}{∂v}+\dfrac{∂w}{∂y}⋅\dfrac{∂y}{∂v}+\dfrac{∂w}{∂z}⋅\dfrac{∂z}{∂v}. This gives us Equation. \end{align*}. \end{align}, Example. Calculate $$\displaystyle ∂z/∂x$$ and $$\displaystyle ∂z/∂y,$$ given $$\displaystyle x^2e^y−yze^x=0.$$. Calculate $$\displaystyle ∂z/∂x,\; ∂z/∂y,\; ∂x/∂u,\; ∂x/∂v,\; ∂y/∂u,$$ and $$\displaystyle ∂y/∂v$$, then use Equation \ref{chain2a} and Equation \ref{chain2b}. \end{equation*}, Theorem. able chain rule helps with change of variable in partial diﬀerential equations, a multivariable analogue of the max/min test helps with optimization, and the multivariable derivative of a scalar-valued function helps to ﬁnd tangent planes and trajectories. That is 2xy, and maybe I should point out that this is w sub x, times dx over dt plus -- Well, w sub y is x squared times dy over dt plus w sub z, which is going to be just one, dz over dt. 1. \end{align*} \], As $$\displaystyle t$$ approaches $$\displaystyle t_0, (x(t),y(t))$$ approaches $$\displaystyle (x(t_0),y(t_0)),$$ so we can rewrite the last product as, \displaystyle \lim_{(x,y)→(x_0,y_0)}\dfrac{(E(x,y)}{\sqrt{(x−x_0)^2+(y−y_0)^2}}\lim_{(x,y)→(x_0,y_0)}(\dfrac{\sqrt{(x−x_0)^2+(y−y_0)^2}}{t−t_0}). Use the chain rule for two parameters with each of the following.(1)\quad F(x,y)=x^2+y^2 where x(u,v)=u \sin v and y(u,v)=u-2v(2)\quad F(x,y)=\ln x y where x(u,v)=e^{u v^2} and y(u,v)=e^{u v}., Exercise. Therefore, this value is finite. Jump to:navigation, search. Let us remind ourselves of how the chain rule works with two dimensional functionals. We compute, \begin{align} \frac{\partial F}{\partial x}& =\frac{\partial F}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial F}{\partial v}\frac{\partial v}{\partial x}+\frac{\partial F}{\partial w}\frac{\partial w}{\partial x} \\ & =\frac{\partial F}{\partial u}(1)+\frac{\partial F}{\partial v}(0)+\frac{\partial F}{\partial w}(-1) \\ & =\frac{\partial F}{\partial u}-\frac{\partial F}{\partial w}. (Chain Rule Involving Two Independent Variables) Suppose z=f(x,y) is a differentiable function at (x,y) and that the partial derivatives of x=x(u,v) and y=y(u,v) exist at (u,v). Then the composite function z=f(x(u,v),y(u,v)) is differentiable at (u,v) with \begin{equation} \frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial … First, take derivatives after direct substitution for , wrtheta = f[ r Cos[theta], r Sin[theta] ] Then try using the Chain Rule directly, The Chain rule of derivatives is a direct consequence of differentiation. For instance, consider the implicit function $$x^2y-xy^3=3.$$ We learned to use the following steps to find $$\frac{dy}{dx}$$: How would we calculate the derivative in these cases? When we put this all together, we get. The generalization of the chain rule to multi-variable functions is rather technical. From EverybodyWiki Bios & Wiki. the chain-rule then boils down to matrix multiplication. Because z=f(x,y) is differentiable, we can write the increment \Delta z in the following form: \begin{equation} \Delta z=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y+\epsilon_1\Delta x+\epsilon_2\Delta y \end{equation} where \epsilon_1\to 0 and \epsilon_2\to 0 as both \Delta x\to 0 and \Delta y\to 0. Dividing by \Delta t\neq 0, we obtain \begin{equation} \frac{\Delta z}{\Delta t}=\frac{\partial z}{\partial x}\frac{\Delta x}{\Delta t}+\frac{\partial z}{\partial y}\frac{\Delta y}{\Delta t}+\epsilon 1\frac{\Delta x}{\Delta t}+\epsilon_2\frac{\Delta y}{\Delta t}. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. If z=e^x\sin y where x=s t^2 and y=s^2t, find \frac{\partial z}{\partial s} and \frac{\partial z}{\partial t}., Solution. However, since x = x(t) and y = y(t) are functions of the single variable t, their derivatives are the standard derivatives of functions of one variable. Active 20 days ago. Calculate $$dz/dt$$ given the following functions. In this equation, both $$\displaystyle f(x)$$ and $$\displaystyle g(x)$$ are functions of one variable. Note, that the sizes of the matrices are automatically of the right. Perform implicit differentiation of a function of two or more variables. 10 Multivariable functions and integrals 10.1 Plots: surface, contour, intensity To understand functions of several variables, start by recalling the ways in which you understand a function f of one variable. Multivariable higher-order chain rule. Calculate $$\displaystyle dy/dx$$ if y is defined implicitly as a function of $$\displaystyle x$$ via the equation $$\displaystyle 3x^2−2xy+y^2+4x−6y−11=0$$. 2 \begingroup I am trying to understand the chain rule under a change of variables. and write out the formulas for the three partial derivatives of $$\displaystyle w$$. Next, we calculate $$\displaystyle ∂w/∂v$$: \[\begin{align*} \dfrac{∂w}{∂v} =\dfrac{∂w}{∂x}⋅\dfrac{∂x}{∂v}+\dfrac{∂w}{∂y}⋅\dfrac{∂y}{∂v}+\dfrac{∂w}{∂z}⋅\dfrac{∂z}{∂v} \\[4pt] =(6x−2y)e^u\cos v−2x(−e^u\sin v)+8z(0), \end{align*}. \end{equation} The chain rule for the case when n=4 and m=2.. The proof of this chain rule is motivated by appealing to a previously proven chain rule with one independent variable. Somehow … Then $$\displaystyle f(x,y)=x^2+3y^2+4y−4.$$ The ellipse $$\displaystyle x^2+3y^2+4y−4=0$$ can then be described by the equation $$\displaystyle f(x,y)=0$$. Chain Rule in Derivatives: The Chain rule is a rule in calculus for differentiating the compositions of two or more functions. Okay, so you know the chain rule from calculus 1, which takes the derivative of a composition of functions. The variables $$\displaystyle x$$ and $$\displaystyle y$$ that disappear in this simplification are often called intermediate variables: they are independent variables for the function $$\displaystyle f$$, but are dependent variables for the variable $$\displaystyle t$$. Khan Academy is a 501(c)(3) nonprofit … In this diagram, the leftmost corner corresponds to $$\displaystyle z=f(x,y)$$. \nonumber\]. \end{align*}\]. \nonumber\]. \end{equation}. Derivative along an explicitly parametrized curve One common application of the multivariate chain rule is when a point varies along acurveorsurfaceandyouneedto・“uretherateofchangeofsomefunctionofthe moving point. For the function f(x,y) where x and y are functions of variable t , we first differentiate the function partially with respect to one variable and then that variable is differentiated with respect to t . Example $$\PageIndex{2}$$: Using the Chain Rule for Two Variables. To use the chain rule, we again need four quantities—$$\displaystyle ∂z/∂x,∂z/dy,dx/dt,$$ and $$\displaystyle dy/dt:$$. \end{equation}, Example. The reason is that, in Note, $$\displaystyle z$$ is ultimately a function of $$\displaystyle t$$ alone, whereas in Note, $$\displaystyle z$$ is a function of both $$\displaystyle u$$ and $$\displaystyle v$$. Chapter 5 uses the results of the three chapters preceding it to prove the Inverse Function Theorem, then the Implicit Function … The Multivariable Chain Rule allows us to compute implicit derivatives easily by just computing two derivatives. Next, we divide both sides by $$\displaystyle t−t_0$$: z(t)−z(t_0)t−t_0=fx(x_0,y_0)(x(t)−x(t_0)t−t_0)+f_y(x_0,y_0)(y(t)−y(t_0)t−t_0)+E(x(t),y(t))t−t_0. Dave4Math » Calculus 3 » Chain Rule for Multivariable Functions. In particular, we will see that there are multiple variants to the chain rule here all depending on how many variables our function is dependent on and how each of those variables can, in turn, be written in terms of different variables. \\ & \hspace{2cm} \left. 0. Therefore, three branches must be emanating from the first node. The online Chain rule derivatives calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. \end{align*}. It is one instance of a chain rule, ... And for that you didn't need multivariable calculus. In this instance, the multivariable chain rule says that df dt = @f @x dx dt + @f @y dy dt. \end{align*}\]. \begin{align*} z =f(x,y)=x^2−3xy+2y^2 \\[4pt] x =x(t)=3\sin2t,y=y(t)=4\cos 2t \end{align*}. Multivariable Chain Rules allow us to differentiate z with respect to any of the variables involved: Let x = x(t) and y = y(t) be differentiable at t and suppose that z = f(x, y) is differentiable at the point (x(t), y(t)). In this multivariable calculus video lesson we will explore the Chain Rule for functions of several variables. Calculate $$\displaystyle ∂w/∂u$$ and $$\displaystyle ∂w/∂v$$ using the following functions: \begin{align*} w =f(x,y,z)=3x^2−2xy+4z^2 \\[4pt] x =x(u,v)=e^u\sin v \\[4pt] y =y(u,v)=e^u\cos v \\[4pt] z =z(u,v)=e^u. Recall from implicit differentiation provides a method for finding $$\displaystyle dy/dx$$ when $$\displaystyle y$$ is defined implicitly as a function of $$\displaystyle x$$. Recall that the chain rule for functions of a single variable gives the rule for differentiating a composite function: if y=f(x) and x=g(t), where f and g are differentiable functions, then y is a a differentiable function of t and \begin{equation} \frac{dy}{d t}=\frac{dy}{dx}\frac{dx}{dt}. Next lesson. 2 \begingroup I am trying to understand the chain rule under a change of variables. Curvature. Theorem. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. \end{equation}, Solution. chain rule. One way of describing the chain rule is to say that derivatives of compositions of differentiable functions may be obtained by linearizing. \[\begin{align*}\dfrac{∂w}{∂t} =\dfrac{∂w}{∂x}\dfrac{∂x}{∂t}+\dfrac{∂w}{∂y}\dfrac{∂y}{∂t} \\[4pt] \dfrac{∂w}{∂u} =\dfrac{∂w}{∂x}\dfrac{∂x}{∂u}+\dfrac{∂w}{∂y}\dfrac{∂y}{∂u} \\[4pt] \dfrac{∂w}{∂v} =\dfrac{∂w}{∂x}\dfrac{∂x}{∂v}+\dfrac{∂w}{∂y}\dfrac{∂y}{∂v} \end{align*}. \end{equation} as desired. To use the chain rule, we need four quantities—$$\displaystyle ∂z/∂x,∂z/∂y,dx/dt$$, and $$\displaystyle dy/dt$$: Now, we substitute each of these into Equation \ref{chain1}: $\dfrac{dz}{dt}=\dfrac{\partial z}{\partial x} \cdot \dfrac{dx}{dt}+\dfrac{\partial z}{\partial y} \cdot \dfrac{dy}{dt}=(8x)(\cos t)+(6y)(−\sin t)=8x\cos t−6y\sin t. \nonumber$, This answer has three variables in it. All we need to do is use the formula for multivariable chain rule. In this article, I cover the chain rule with several independent variables. One way of describing the chain rule is to say that derivatives of compositions of differentiable functions may be obtained by linearizing. +\frac{\partial }{\partial s}\left(\frac{\partial u}{\partial y}\right)e^s \sin t +\frac{\partial u}{\partial x}\left(-e^s \cos t\right)+\frac{\partial }{\partial t}\left(\frac{\partial u}{\partial x}\right)\left(-e^s \sin t\right) \right. Suppose that $$\displaystyle x=g(t)$$ and $$\displaystyle y=h(t)$$ are differentiable functions of $$\displaystyle t$$ and $$\displaystyle z=f(x,y)$$ is a differentiable function of $$\displaystyle x$$ and $$\displaystyle y$$. Notation, domain, and range; Parameterized surfaces; Quadratic surfaces; Surfaces; Surfaces in other coordinate systems; Traces, contours, and level sets; Differentiation of multivariable functions. b. The chain rule gives, \begin{align} \frac{d z}{d t} &=\frac{\partial z}{\partial x}\frac{d x}{d t}+\frac{\partial z}{\partial y}\frac{d y}{d t} \\ & =\left(2e^t\sin t+3 \text{sin t}^4t\right)e^t +\left(e^{2t}+12e^t\sin ^3t\right) \cos t. \end{align} as desired., Exercise. Partial Derivative / Multivariable Chain Rule Notation. \end{align*}\], The left-hand side of this equation is equal to $$\displaystyle dz/dt$$, which leads to, $\dfrac{dz}{dt}=f_x(x_0,y_0)\dfrac{dx}{dt}+f_y(x_0,y_0)\dfrac{dy}{dt}+\lim_{t→t_0}\dfrac{E(x(t),y(t))}{t−t_0}. State the chain rules for one or two independent variables. \end{equation}, Solution. $$\displaystyle \dfrac{∂z}{∂u}=0,\dfrac{∂z}{∂v}=\dfrac{−21}{(3\sin 3v+\cos 3v)^2}$$. Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables. \\ & \hspace{2cm} \left. In this case the chain rule says: DF (t) = (Df)(g(t)) . Write out the chain rule for the case for the case when n=4 and m=2 where w=f(x,y,z,t), x=x(u,v), y=y(u,v), z=z(u,v), and t(u,v)., Solution. The chain rule consists of partial derivatives . This equation implicitly defines $$\displaystyle y$$ as a function of $$\displaystyle x$$. +\frac{\partial u}{\partial y}\left(-e^s \sin t\right) +\frac{\partial ^2 u}{\partial x \partial y}\left(-e^{2s} \cos t \sin t\right) +\frac{\partial ^2 u}{\partial y^2}e^{2s} \cos ^2 t\right] \\ & =e^{-2s}\left[\frac{ \partial ^2 u}{\partial x^2}e^{2s} \cos ^2 t +\frac{ \partial ^2 u}{\partial y^2}e^{2s} \sin ^2 t +\frac{ \partial ^2 u}{\partial x^2}\left(e^{2s} \sin ^2 t\right) +\frac{ \partial ^2 u}{\partial y^2}e^{2s} \cos ^2 t\right] \\ & =\frac{ \partial ^2u}{\partial x^2}+\frac{ \partial ^2u}{\partial y^2}. 8.2 Chain Rule For functions of one variable, the chain rule allows you to di erentiate with respect to still another variable: ya function of xand a function of tallows dy dt = dy dx dx dt (8:3) You can derive this simply from the de nition of a derivative. Then, $$\displaystyle z=f(g(u,v),h(u,v))$$ is a differentiable function of $$\displaystyle u$$ and $$\displaystyle v$$, and, \[\dfrac{∂z}{∂u}=\dfrac{∂z}{∂x}\dfrac{∂x}{∂u}+\dfrac{∂z}{∂y}\dfrac{∂y}{∂u} \label{chain2a}$, $\dfrac{∂z}{∂v}=\dfrac{∂z}{∂x}\dfrac{∂x}{∂v}+\dfrac{∂z}{∂y}\dfrac{∂y}{∂v}. In the real world, it is very difficult to explain behavior as a function of only one variable, and economics is no different. One way of describing the chain rule is to say that derivatives of compositions of differentiable functions may be obtained by linearizing. $$\displaystyle \dfrac{∂w}{∂v}=\dfrac{15−33\sin 3v+6\cos 3v}{(3+2\cos 3v−\sin 3v)^2}$$, Example $$\PageIndex{4}$$: Drawing a Tree Diagram, \[ w=f(x,y,z),x=x(t,u,v),y=y(t,u,v),z=z(t,u,v) \nonumber$. (Dg)(t), or using functions. If $f$ is differentiable and $z=u+f\left(u^2v^2\right)$, show that \begin{equation} u\frac{\partial z}{\partial u}-v\frac{\partial z}{\partial v}=u. Chapter 5 … \end{equation}. }\) Find $$\ds \frac{dz}{dt}$$ using the Chain Rule. As Preview Activity 10.3.1 suggests, the following version of the Chain Rule holds in general. Lästid: ~15 min Visa alla steg. Implicit function theorem notation question. Multivariable Chain Rule SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 13.5 of the rec-ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. Applying the chain rule we obtain \begin{align} \frac{\partial z}{\partial s} & =\frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z} {\partial y}\frac{\partial y}{\partial s} \\ & =\left(e^x\sin y\right)\left(t^2\right)+\left(e^x\cos y\right)( s t) \\ & =t^2e^{s t^2}\sin \left(s^2 t\right)+2s t e^{s t^2}\cos \left(s^2t\right) \end{align} and \begin{align} \frac{\partial z}{\partial t} &=\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t} \\ & =\left(e^x\sin y\right)(2 s t)+\left(e^x\cos y\right)\left(2 s^2\right) \\ & =2 s t e^{s t^2}\sin \left(s^2 t\right)+s^2 e^{s t^2}\cos \left(s^2t\right). Pick up a machine learning paper or the documentation of a library such as PyTorch and calculus comes screeching back into your life like distant relatives around the holidays. \begin{align} \frac{df}{dt} = \frac{df}{dx}\frac{dx}{dt} + \frac{df}{dy}\frac{dy}{dt} \end{align} I found multiple derivation of this results online using differentials and mean value theorem, but they don't look like rigorous to me. New Resources. To implement the chain rule for two variables, we need six partial derivatives—$$\displaystyle ∂z/∂x,\; ∂z/∂y,\; ∂x/∂u,\; ∂x/∂v,\; ∂y/∂u,$$ and $$\displaystyle ∂y/∂v$$: \begin{align*} \dfrac{∂z}{∂x} =6x−2y \dfrac{∂z}{∂y}=−2x+2y \\[4pt] \displaystyle \dfrac{∂x}{∂u} =3 \dfrac{∂x}{∂v}=2 \\[4pt] \dfrac{∂y}{∂u} =4 \dfrac{∂y}{∂v}=−1. Multivariable Chain Formula Given function f with variables x, y and z and x, y and z being functions of t, the derivative of f with respect to t is given by by the multivariable chain rule which is a sum of the product of … Solution A: We'll use theformula usingmatrices of partial derivatives:Dh(t)=Df(g(t))Dg(… Collection of Multivariable Chain Rule exercises and solutions, Suitable for students of all degrees and levels and will help you pass the Calculus test successfully. Multivariate Chain Rule. able chain rule helps with change of variable in partial diﬀerential equations, a multivariable analogue of the max/min test helps with optimization, and the multivariable derivative of a scalar-valued function helps to ﬁnd tangent planes and trajectories. Well, the chain rule tells us that dw/dt is, we start with partial w over partial x, well, what is that? Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. The Cauchy-Riemann equations are \begin{equation} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \qquad \text{and} \qquad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\end{equation} where u=u(x,y) and v=v(x,y). Show that if x and y are expressed in terms of polar coordinates, the Cauchy-Riemann equations become \begin{equation} \frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta } \qquad \text{and} \qquad \frac{\partial v}{\partial r}=\frac{-1}{r}\frac{\partial u}{\partial \theta }. In the section we extend the idea of the chain rule to functions of several variables. The Multivariable Chain Rule is used to differentiate functions with inputs of multiple variables. Viewed 136 times 5. \[\begin{align*} \dfrac{∂w}{∂t} =\dfrac{∂w}{∂x}\dfrac{∂x}{∂t}+\dfrac{∂w}{∂y}\dfrac{∂y}{∂t}+\dfrac{∂w}{∂z}\dfrac{∂z}{∂t} \\[4pt] \dfrac{∂w}{∂u} =\dfrac{∂w}{∂x}\dfrac{∂x}{∂u}+\dfrac{∂w}{∂y}\dfrac{∂y}{∂u}+\dfrac{∂w}{∂z}\dfrac{∂z}{∂u} \\[4pt] \dfrac{∂w}{∂v} =\dfrac{∂w}{∂x}\dfrac{∂x}{∂v}+\dfrac{∂w}{∂y}\dfrac{∂y}{∂v}+\dfrac{∂w}{∂z}\dfrac{∂z}{∂v}. \\ & \hspace{2cm} \left. 12.5: The Multivariable Chain Rule. Last, each of the branches on the far right has a label that represents the path traveled to reach that branch. Equation \ref{implicitdiff1} can be derived in a similar fashion. Computing the derivatives shows df dt = (2x) (2t) + (2y) (4t3). 11 Partial derivatives and multivariable chain rule 11.1 Basic deﬁntions and the Increment Theorem One thing I would like to point out is that you’ve been taking partial derivatives all your calculus-life. \end{align*}. All rights reserved. If $F(u,v,w)$ is differentiable where $u=x-y,$ $v=y-z,$ and $w=z-x,$ then find  \frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}. which is the same result obtained by the earlier use of implicit differentiation. Multivariable Calculus The chain rule. \end{align*}\]. 2. Let $w=f(t)$ be a differentiable function of $t$, where $t =\left(x^2+y^2 +z^2\right)^{1/2}.$ Show that \begin{equation} \left( \frac{d w}{d t} \right)^2=\left( \frac{\partial w}{\partial x} \right)^2+\left( \frac{\partial w}{\partial y} \right)^2+\left(\frac{\partial w}{\partial z} \right)^2.\end{equation}, Exercise. Multivariable Chain Rule. The Chain Rule with One Independent Variable, The Chain Rule with Two Independent Variables, The Chain Rule with Several Independent Variables, Choose your video style (lightboard, screencast, or markerboard), chain rule for functions of a single variable, Derivatives and Integrals of Vector Functions (and Tangent Vectors) [Video], Vector Functions and Space Curves (Calculus in 3D) [Video], Probability Density Functions (Applications of Integrals), Conservative Vector Fields and Independence of Path, Jacobian (Change of Variables in Multiple Integrals), Triple Integrals in Cylindrical and Spherical Coordinates. The method involves differentiating both sides of the equation defining the function with respect to $$\displaystyle x$$, then solving for $$\displaystyle dy/dx.$$ Partial derivatives provide an alternative to this method. The first term in the equation is $$\displaystyle \dfrac{∂f}{∂x} \cdot \dfrac{dx}{dt}$$ and the second term is $$\displaystyle \dfrac{∂f}{∂y}⋅\dfrac{dy}{dt}$$. 2. \end{align*}\]. \end{equation}, Solution. +\frac{ \partial ^2 u}{\partial x^2}\left(e^{2s} \sin ^2 t\right) +\frac{ \partial ^2 u}{\partial x \partial y}\left(-e^{2s} \cos t \sin t\right) \right. Then we take the limit as $$\displaystyle t$$ approaches $$\displaystyle t_0$$: \[\begin{align*} \lim_{t→t_0}\dfrac{z(t)−z(t_0)}{t−t_0} = f_x(x_0,y_0)\lim_{t→t_0} \left (\dfrac{x(t)−x(t_0)}{t−t_0} \right) \\[4pt] +f_y(x_0,y_0)\lim_{t→t_0}\left (\dfrac{y(t)−y(t_0)}{t−t_0}\right)\\[4pt] +\lim_{t→t_0}\dfrac{E(x(t),y(t))}{t−t_0}. surface (x,y,z)=f(u,v). (Chain Rule Involving Several Independent Variable) If $w=f\left(x_1,\ldots,x_n\right)$ is a differentiable function of the $n$ variables $x_1,…,x_n$ which in turn are differentiable functions of $m$ parameters $t_1,…,t_m$ then the composite function is differentiable and \begin{equation} \frac{\partial w}{\partial t_1}=\sum_{k=1}^n \frac{\partial w}{\partial x_k}\frac{\partial x_k}{\partial t_1}, \quad … \quad , \frac{\partial w}{\partial t_m}=\sum_{k=1}^n \frac{\partial w}{\partial x_k}\frac{\partial x_k}{\partial t_m}.\end{equation}, Example. What is the equation of the tangent line to the graph of this curve at point $$\displaystyle (2,1)$$? \begin{align} & \left.\frac{\partial s}{\partial x_1}\right|_{t=\pi } =\left.\frac{-\left(x_2-x_1\right)}{\sqrt{\left(x_2-x_1\right){}^2+\left(y_2-y_1\right){}^2}} \right|_{t=\pi}=\frac{-2}{\sqrt{13}} \\ & \left.\frac{\partial s}{\partial y_1}\right|_{t=\pi } =\left.\frac{-\left(y_2-y_1\right)}{\sqrt{\left(x_2-x_1\right){}^2+\left(y_2-y_1\right){}^2}}\right|_{t=\pi} = \frac{-3}{\sqrt{13}} \\ & \left.\frac{\partial s}{\partial x_2}\right|_{t=\pi } =\left.\frac{\left(x_2-x_1\right)}{\sqrt{\left(x_2-x_1\right){}^2+\left(y_2-y_1\right){}^2}}\right|{t=\pi}=\frac{2}{\sqrt{13}} \\ & \left.\frac{\partial s}{\partial y_2}\right|_{t=\pi } =\left.\frac{\left(y_2-y_1\right)}{\sqrt{\left(x_2-x_1\right){}^2+\left(y_2-y_1\right){}^2}}\right|_{t=\pi}=\frac{3}{\sqrt{13}} \end{align} When $t=\pi ,$ the derivatives of $x_1,$ $y_1,$ $x_2,$ and $y_2$ are \begin{align} & \left.\frac{d x_1}{dt}\right|_{t=\pi }=-2 \sin t|{t=\pi }=0 & & \left.\frac{d y_1}{dt}\right|_{t=\pi }=3 \cos t|{t=\pi }=-3 \\ & \left.\frac{d x_2}{dt}\right|_{t=\pi }=8 \cos 2t|{t=\pi }=8 & & \left.\frac{d y_2}{dt}\right|_{t=\pi }=-6 \sin 2t|{t=\pi }=0 \end{align} So using the chain rule \begin{equation} \frac{d s}{d t} =\frac{\partial s}{\partial x_1}\frac{d x_1}{d t}+\frac{\partial s}{\partial y_1}\frac{d y_1}{d t}+\frac{\partial s}{\partial x_2}\frac{d x_2}{d t}+\frac{\partial s}{\partial y_2}\frac{d y_2}{d t} \end{equation} When $t=\pi$, we find that the distance is changing at a rate of \begin{equation*} \left.\frac{d s}{d t} \right|_{t=\pi} =\left(\frac{-2}{\sqrt{13}}\right)(0)+\left(\frac{-3}{\sqrt{13}}\right)(-3)+\left(\frac{2}{\sqrt{13}}\right)(8)+\left(\frac{3}{\sqrt{13}}\right)(0) =\frac{25}{\sqrt{13}}. 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