# Section 5: The Chain Rule

### Instructions

• First, you should watch the concepts videos below explaining the topics in the section.
• Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
• Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
• When you have finished the material below, you can start on the next section or return to the main several variable calculus page.

### Concepts

• The Chain Rule for functions of more than one variable
• Related rates

If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.

### Exercises

Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. If $$z=\ln(x^2+y^2)$$, $$x=1+e^{6t}$$, $$y=\sec^2(5t)$$, find $$\dfrac{dz}{dt}.$$

$$\dfrac{dz}{dt}=\dfrac{2x}{x^2+y^2}\cdot6e^{6t}+\dfrac{2y}{x^2+y^2}\cdot 2\sec(5t)\cdot \sec(5t)\tan(5t)$$

If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.

2. If $$z=x^2-3x^2y^3$$, $$x=se^t$$, $$y=se^{-t}$$, find $$\dfrac{\partial z}{\partial s}$$ and $$\dfrac{\partial z}{\partial t}.$$

$$\dfrac{dz}{ds}=\left(2x-6xy^3\right)e^t-9x^2y^2e^{-t}$$

$$\dfrac{dz}{dt}=\left(2x-6xy^3\right)\left(5e^t\right)+45x^2y^2e^{-t}$$

If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.

3. If $$u=x^4y+y^2z^3$$, $$x=rse^t$$, $$y=rs^2e^{-t}$$, $$z=r^2s\sin t$$, find $$\displaystyle{\frac{\partial u}{\partial t}}$$ when $$r=2$$, $$s=3$$ and $$t=0.$$

$$\left.\dfrac{\partial u}{\partial t}\right|_{r=2,s=3,t=0}=69,984$$

If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.

4. The radius of a circular cone is increasing at a rate of 1.8 inches per second, while the height is decreasing at a rate of 2.5 inches per second. At what rate is the volume of the cone changing when the radius is 120 inches and the height is 140 inches?

$$8160\pi \frac{\text{in}^3}{\text{sec}}$$

If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. What are the requirements in order to use the chain rule? Write your own question that uses the chain rule where $$w=f(x,y,z)$$, where $$x$$, $$y$$ and $$z$$ are all functions of $$u$$ and $$v$$. Draw the tree diagram and solve the problem.
2. Write your own related rates word problem where two or more quantities are changing with respect to time and find the rate at which a third quantity is changing by using a formula that relates these quantities.