# 6.4: The Chain Rule

- Page ID
- 106371

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**Here is a good review of Compositions of Functions, if needed:**

### Preview Video:

This optional summary video** **from the textbook author might be helpful to use as a preview. Other, more detailed, supplemental videos for this section are posted at the end of the text.

## Supplemental Videos

**Example of the Chain rule - polynomials**

**Example of Chain Rule with Radicals**

**Example of Chain Rule with Trig Functions**

**Example of Chain Rule with Exponential Functions **

**Examples mixing rules**

**Example of Chain rule when given the graphs **

## Homework Exercises 6.4

**WeBWorK Problems:**

**Written Problems:**

**1. ** If a tank holds 4000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli's Law gives the volume *V* of water remaining in the tank after *t *minutes as

\(V=4000(1-\frac{t}{40})^2\), \(0\leq t \leq 40\).

**a. **Find the rate at which the water is draining out of the tank after *t* minutes.

**b. **Find the rate at which the water is draining out of the tank after 10 minutes.

**2. **A mass attached to a vertical spring has position function given by \(s(t)=4 sin(2t)\) where *t* is measured in seconds and *s* in inches.

**a. **Find the velocity at *t* = 2.

**b. **Find the acceleration at *t* = 2.

**3. **The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose that the equation of motion of a point on a spring is \(s(t)=2e^{-1.5t} sin(2\pi t)\), where *s *is measured in centimeters and *t* in seconds. Find the velocity after *t* seconds.

**4. **

**5. ** If a spherical tank of radius 4 feet has ℎ feet of water present in the tank, then the volume of water in the tank is given by the formula \(V=\frac{\pi}{3}h^2(12-h)\).

**a. **At what instantaneous rate is the volume of water in the tank changing with respect to the * height* of the water at the instant ℎ=1? What are the units on this quantity?

**b. **Now suppose that the height of water in the tank is being regulated by an inflow and outflow (e.g., a faucet and a drain) so that the height of the water at time *t* is given by the rule \(h(t)= sin (\pi t)+1\), where *t *is measured in hours (and ℎ is still measured in feet). At what rate is the height of the water changing with respect to time at the instant *t *=2?

**c. **Continuing under the assumptions in (b), at what instantaneous rate is the volume of water in the tank changing with respect to * time* at the instant

*t*=2?

**d. **What are the main differences between the rates found in (a) and (c)? Include a discussion of the relevant units.