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Chapter 3: Problem 44

Use the General Power Rule where appropriate to find the derivative of the following functions. $$f(x)=\left(2^{x}+1\right)^{\pi}$$

Short Answer

Expert verified
Answer: The derivative of the function is \(f'(x) = \pi(2^x + 1)^{\pi - 1} \cdot 2^x \ln(2)\).

Step by step solution

01

Identify the inner and outer functions

The given function can be represented as \(f(x) = (g(x))^{\pi}\), where \(g(x) = 2^x + 1\). So, the inner function is \(g(x) = 2^x + 1\) and the outer function is \(f(g(x)) = g(x)^{\pi}\).
02

Find the derivative of the inner function

To find \(g'(x)\), the derivative of \(g(x)\) with respect to \(x\), we use the Exponential Rule. The Exponential Rule states that the derivative of \(a^x\) is \(a^x \ln(a)\). $$g'(x) = \frac{d}{dx}(2^x + 1) = 2^x \ln(2) + 0 = 2^x \ln(2)$$
03

Apply the General Power Rule

The General Power Rule combines the Chain Rule and the Exponential Rule. If \(f(x) = (g(x))^{\pi}\), then \(f'(x) = \pi \cdot (g(x))^{\pi - 1} \cdot g'(x)\). We substitute \(g(x) = 2^x + 1\) and its derivative, \(g'(x) = 2^x \ln(2)\), obtained in the previous step. $$f'(x) = \pi \cdot ((2^x + 1)^{\pi - 1}) \cdot (2^x \ln(2))$$
04

Simplify the derivative

Now we simplify the expression for the derivative: $$f'(x) = \pi(2^x + 1)^{\pi - 1} \cdot 2^x \ln(2)$$ This is the derivative of the given function using the General Power Rule.

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Most popular questions from this chapter

Special Product Rule In general, the derivative of a product is not the product of the derivatives. Find nonconstant functions \(f\) and \(g\) such that the derivative of \(f g\) equals \(f^{\prime} g^{\prime}\).

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(f(x) g(x))\right|_{x=1}$$

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\) $$f(x)=x^{2}+1, \text { for } x \geq 0 ;(5,2)$$

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x)}{(x+2)}\right]\right|_{x=4}$$

Derivatives from tangent lines Suppose the line tangent to the graph of \(f\) at \(x=2\) is \(y=4 x+1\) and suppose \(y=3 x-2\) is the line tangent to the graph of \(g\) at \(x=2 .\) Find an equation of the line tangent to the following curves at \(x=2\) a. \(y=f(x) g(x)\) b. \(y=\frac{f(x)}{g(x)}\)
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