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Problem 10

# Calculate the derivatives of the functions in Exercises 1-46. HINT [See Example 1.] $$f(x)=(x+7)^{-2}$$

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The derivative of the function $$f(x) = (x+7)^{-2}$$ is $$f'(x) = -2(x+7)^{-3}$$.
See the step by step solution

## Step 1: Rewrite the function

Rewrite the function as: $$f(x) = (x+7)^{-2}$$.

## Step 2: Identify the outer and inner functions

In this case, the outer function is $$g(x) = x^{-2}$$, and the inner function is $$h(x) = x+7$$. Our goal is to find the derivative $$f'(x) = \frac{d}{dx} ((x+7)^{-2})$$ by applying the chain rule.

## Step 3: Apply the Chain Rule for Derivatives

Using the chain rule, which states $$\frac{d}{dx} (g(h(x))) = g'(h(x))h'(x)$$, we will find the derivative of the outer function and the inner function: Outer function derivative: $$g'(x) = \frac{d}{dx} (x^{-2}) = -2x^{-3}$$. Inner function derivative: $$h'(x) = \frac{d}{dx}(x+7) = 1$$.

## Step 4: Substitute and Simplify

Substitute the derivatives we found in step 3 into the chain rule formula: $$f'(x) = g'(h(x))h'(x) = (-2(x+7)^{-3})(1) = -2(x+7)^{-3}$$. The derivative of the given function is: $$f'(x) = -2(x+7)^{-3}$$.

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