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Chapter 3: Problem 42

Use the Chain Rule to find the derivative of the following functions. $$y=\left(1-e^{x}\right)^{4}$$

Short Answer

Expert verified
Question: Find the derivative of the function $$y=(1-e^x)^4$$ using the Chain Rule. Answer: The derivative of the function $$y=(1-e^x)^4$$ is $$\frac{dy}{dx} = -4e^x(1-e^x)^3$$.

Step by step solution

01

Identify the functions

We can decompose the given function into two parts: the outer function $$u^4$$, and the inner function $$1-e^x$$, where $$u = 1-e^x$$.
02

Find the derivative of the outer function

To find the derivative of the outer function $$u^4$$, treat u as a variable, and differentiate with respect to u: $$\frac{d}{du}(u^4) = 4u^3$$
03

Find the derivative of the inner function

The inner function is $$1-e^x$$. Differentiate with respect to x: $$\frac{d}{dx}(1-e^x) = -e^x$$
04

Apply the Chain Rule

Now, we'll apply the Chain Rule by multiplying the derivatives found in steps 2 and 3: $$\frac{dy}{dx} = \frac{d}{du}(u^4) \ \frac{du}{dx}$$ Substitute the derivatives found in steps 2 and 3: $$\frac{dy}{dx} = (4u^3)(-e^x)$$
05

Substitute the inner function back in

Finally, substitute the inner function $$u=1-e^x$$ back into the derivative: $$\frac{dy}{dx} = 4(1-e^x)^3(-e^x)$$ So, the derivative of the given function is: $$\frac{dy}{dx} = -4e^x(1-e^x)^3$$

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