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Chapter 3: Problem 56

Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. $$y=\left(\frac{e^{x}}{x+1}\right)^{8}$$

Short Answer

Expert verified
Answer: The derivative of the function $$y=\left(\frac{e^{x}}{x+1}\right)^{8}$$ is: $$(y)'= \frac{8e^{x}x\left(\frac{e^{x}}{x+1}\right)^{7}}{(x+1)^2}$$.

Step by step solution

01

Identify the outer function and its derivative

The outer function is the one raised to the power of 8, and the inner function is the fraction. The derivative of the outer function, which is a power function, can be found using the power rule: \((x^n)'= nx^{n-1}\). So in our case, the derivative of the outer function is: $$8\left(\frac{e^{x}}{x+1}\right)^{7}$$.
02

Identify the inner function and apply the Quotient Rule

The inner function is the fraction itself: $$\frac{e^{x}}{x+1}$$. To find its derivative, we'll apply the Quotient Rule, which is: $$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$. In our case, $$u = e^x$$ and $$v = x+1$$.
03

Differentiate u and v

Now, let's differentiate both functions $$u$$ and $$v$$. Differentiating $$u=e^x$$, we get: $$u'=e^x$$ by using the Exponential Rule. Differentiating $$v=x+1$$, we get: $$v'=1$$ by using the power rule for the linear function.
04

Apply the Quotient Rule

With the derivatives found, let's apply the Quotient Rule to find the derivative of the inner function: $$\left(\frac{e^x}{x+1}\right)' = \frac{e^x (x+1) - e^x(1)}{(x+1)^2}$$. Simplify it: $$\frac{e^x x}{(x+1)^2}$$.
05

Combine the derivatives using the Chain Rule

Now, let's apply the Chain Rule to combine the derivatives of the outer and inner functions: $$(y)'= 8\left(\frac{e^{x}}{x+1}\right)^{7} \cdot \frac{e^x x}{(x+1)^2}$$.
06

Simplify the expression

Finally, we can simplify the expression to get the final derivative: $$(y)'= \frac{8e^{x}x\left(\frac{e^{x}}{x+1}\right)^{7}}{(x+1)^2}$$. The derivative of the function $$y=\left(\frac{e^{x}}{x+1}\right)^{8}$$ is: $$(y)'= \frac{8e^{x}x\left(\frac{e^{x}}{x+1}\right)^{7}}{(x+1)^2}$$.

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