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Chapter 3: Problem 37

Find the derivative of the following functions. $$f(x)=3 x^{-9}$$

Short Answer

Expert verified
Answer: The derivative of the function \(f(x) = 3x^{-9}\) is \(f'(x) = -27x^{-10}\).

Step by step solution

01

Identify the power rule for differentiation

We will use the power rule, which states that for any function of the form \(g(x) = x^n\), the derivative is \(g'(x) = nx^{n-1}\).
02

Apply the power rule to the function

Now, let's apply the power rule to the function \(f(x) = 3x^{-9}\): $$ f'(x) = \frac{d}{dx}(3x^{-9}). $$
03

Differentiate the function

Here, we simply use the power rule to find the derivative: $$ f'(x) = 3 \cdot (-9)x^{-9-1}. $$
04

Simplify the derivative

Finally, let's simplify the derivative by combining the coefficients and combining the exponents: $$ f'(x) = -27x^{-10}. $$ The derivative of the function \(f(x) = 3x^{-9}\) is \(f'(x) = -27x^{-10}\).

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