Section 3.5 Chain Rules

3.5 Chain Rules

In this section, we learn the chain rule for the multi-variables function. Recall in calculus I, given a function $y=f(g(t))=f(x),$ with $x=g(t)$, then 

$$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}=f^{\prime }(g(t))g^{\prime }(t)$$

 if both $y$ and $x$ are differentiable. We use the similar approach for multi-variables but we need partial derivative because we can only take derivative at one direction at a time.

Theorem: Chain Rule for One Independent Variable

Suppose that $x=g(t)$ and $y=h(t)$ are differentiable functions of $t$ and $z=f(x,y)$ is a differentiable function of $x$ and $y$. Then $z=f(g(t),h(t))$ is a differentiable function of $t$ and 

$$\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$$

where the ordinary derivatives are evaluated at $t$ and the partial derivatives are evaluated at $(x,y)$.

Idea of the proof using the following equation:

$$z(t)=f(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)+E(x,y),$$

Example 1: Find $\frac{dz}{dt}$, where $z=f(x,y)=x^{2}y-x{y}^2+2x+y$, $x=x(t)=\text{sin}(t)$, and $y=y(t)=\text{cos}(t)$. 

Exercise 1: Find $\frac{dz}{dt}$, where $z=f(x,y)=2x^2+x^{2}y-y^3$, $x=x(t)=\text{tan}(t)$, and $y=y(t)=\text{sec}(t)$. 

Example 2: Find $\frac{dz}{dt}$, where $z=f(x,y)=\sqrt{x^2-y^2}$, $x=x(t)=e^{2t}$, and $y=y(t)=e^{t^2}$. 

Exercise 2: Find $\frac{dz}{dt}$, where $z=f(x,y)=\sqrt{x^2+y^2}$, $x=x(t)=\text{ln}(t)$, and $y=y(t)=e^{-t}$. 

Theorem: Chain Rule for Two Independent Variables

Suppose $x=g(u,v)$ and $y=h(u,v)$ are differentiable functions of $u$ and $v$, and $z=f(x,y)$ is a differentiable function of $x$ and $y$. Then, $z=f(g(u,v),h(u,v))$ is a differentiable function of $u$ and $v$. Moreover,

$$\frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u}$$

and 

$$\frac{\partial z}{\partial v}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial v}.$$

Example 3: Find $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\partial v},$, where $z=f(x,y)=3x^3-x{y}^2-y^3$, $x=x(u,v)=u^2+v$, and $y=y(u,v)=\text{ln}(u)+v$. 

Exercise 3: Find $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\partial v},$where $z=f(x,y)=x^2+xy+y^2$, $x=x(u,v)=3u+4v$, and $y=y(u,v)=2u-v$. 

Theorem Implicit Differentiation of a Function of Two or More Variables

Suppose the equation $f(x,y)=0$ defines $y$ implicitly as a differentiable function of $x$.  Then

$$\frac{dy}{dx}=-\frac{f_x(x,y)}{f_y(x,y)}$$

provided $f_y(x,y)\ne 0$. If the equation $f(x,y,z)=0$ defines $z$ implicitly as a differentiable function of $x$ and $y$, then 

$$\frac{\partial z}{\partial x}=-\frac{f_x(x,y,z)}{f_z(x,y,z)}$$

and 

$$\frac{\partial z}{\partial y}=-\frac{f_y(x,y,z)}{f_z(x,y,z)}$$

as long as $f_z(x,y,z)\ne 0$. 

Example 4: Calculate $\frac{dy}{dx}$ if $y$ is defined implicitly as a function of $x$ via the equation $3x^2+2xy+y^2+5y-11=0$. What is the equation of the tangent line to the graph of this curve at point $(1,1)$?

Exercise 4: Calculate $\frac{dy}{dx}$ if $y$ is defined implicitly as a function of $x$ via the equation $x^2-3xy+2y^2+5x-1=0$. What is the equation of the tangent line to the graph of this curve at point $(-1,1)$?

Example 5: Calculate $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$, given $x^2{e}^y-yz{e}^x=0$.

Exercise 5: Calculate $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$, given $y{e}^x+\text{ln}(x)z^2=0$.

Example 6: The volume of a right circular cylinder is given by $V(x,y)=\pi x^{2}y$, where $x$ is the radius of the cylinder and $y$ is the cylinder height. Suppose $x$ and $y$ are functions of $t$ given by $x=\frac{1}{3}t$ and $y=t$ so that $x$ and $y$ are both increasing with time. How fast is the volume increasing when $x=3$ and $y=4$?

Example 7: A closed box is in the shape of a rectangular solid with dimensions $x$, $y$, and $z$. (Dimensions are in inches.) Suppose each dimension is changing at the rate of $0.4$ in./min. Find the rate of change of the total surface area of the box when $x=2$ in., $y=2$ in., and $z=1$ in.

Group work:

1. The volume of a right circular cone is given by $V(x,y)=\frac{1}{3}\pi x^{2}y$, where $x$ is the radius of the base and $y$ is the height of the cone. Suppose $x$ and $y$ are functions of $t$ given by $x=\frac{1}{2}t$ and $y=2t$ so that $x$ and $y$ are both increasing with time. How fast is the volume increasing when $x=2$ and $y=8$?

2. A closed box is in the shape of a rectangular solid with dimensions $x$, $y$, and $z$. (Dimensions are in inches.) Suppose each dimension is changing at the rate of $0.3$ in./min. Find the rate of change of the total surface area of the box when $x=1$ in., $y=2$ in., and $z=3$ in.

3. The $x$ and $y$ components of a fluid moving in two dimensions are given by the following functions: $u(x,y)=2y$ and $v(x,y)=-2x$; $x\ge 0;y\ge 0$. The speed of the fluid at the point $(x,y)$ is $s(x,y)=u(x,y{)}^2+v(x,y{)}^2$. Find $\frac{\partial s}{\partial x}$ and $\frac{\partial s}{\partial y}$ using the chain rule.

4. Let $u=u(x,y,z)$, where $x=x(w,t)$, $y=y(w,t)$, $z=z(w,t)$, $w=w(r,s)$, and $t=t(r,s)$. Use a tree diagram and the chain rule to find an expression for $\frac{\partial u}{\partial r}$.