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Chapter 6: Problem 47

Find the derivatives of the following functions. $$f(x)=\cosh ^{-1} 4 x$$

Short Answer

Expert verified
Question: Find the derivative of the function \(f(x) = \cosh^{-1}(4x)\). Solution: Using the chain rule and the derivative formula for the inverse hyperbolic cosine function, we find that the derivative of the given function is: $$\frac{d}{dx}f(x) = \frac{4}{\sqrt{16x^2 - 1}}$$

Step by step solution

01

Recall the derivative formula for the inverse hyperbolic cosine

Recall that the derivative of the inverse hyperbolic cosine function with respect to \(x\) is given by: $$\frac{d}{dx}\cosh^{-1}(x) = \frac{1}{\sqrt{x^2 - 1}}$$ We will need this formula to find the derivative of \(f(x)\).
02

Apply the chain rule

Since our function \(f(x)\) is a composition of two functions (\(\cosh^{-1}(u)\) with \(u=4x\)), we need to use the chain rule to find its derivative. The chain rule states that: $$\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$$ In our case, \(f(u) = \cosh^{-1}(u)\) and \(g(x) = 4x\).
03

Compute the derivatives of each function

First, we need to compute the derivative of \(f(u) = \cosh^{-1}(u)\) with respect to \(u\) using the formula mentioned in Step 1: $$\frac{d}{du}\cosh^{-1}(u) = \frac{1}{\sqrt{u^2 - 1}}$$ Next, we need to compute the derivative of \(g(x) = 4x\) with respect to \(x\): $$\frac{d}{dx}(4x) = 4$$
04

Apply the chain rule

Now we can apply the chain rule to find the derivative of \(f(x) = \cosh^{-1}(4x)\): $$\frac{d}{dx}\cosh^{-1}(4x) = \frac{d}{du}\cosh^{-1}(4x) \cdot \frac{d}{dx}(4x)$$ Substituting the derivatives of each function from Step 3, we get: $$\frac{d}{dx}\cosh^{-1}(4x)= \frac{1}{\sqrt{(4x)^2 - 1}}\cdot 4$$
05

Simplify the derivative

Simplify the expression to get the final answer for the derivative of \(f(x)\): $$\frac{d}{dx}\cosh^{-1}(4x) = \frac{4}{\sqrt{(4x)^2 - 1}} = \frac{4}{\sqrt{16x^2 - 1}}$$ The derivative of the given function is: $$\frac{d}{dx}f(x) = \frac{4}{\sqrt{16x^2 - 1}}$$

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