Question

1. submit answer practice similar attempt 1: 10 attempts remaining. calculate the derivative of the function. ( g(x) = left(7x^2 + x + 7

ight)^{-8} ) ( g(x) = )

Explanation:

Step1: Identify the outer and inner functions

The function \( g(x) = (7x^2 + x + 7)^{-8} \) is a composite function. Let \( u = 7x^2 + x + 7 \) (inner function) and \( y = u^{-8} \) (outer function).

Step2: Differentiate the outer function

Using the power rule, if \( y = u^n \), then \( y' = n u^{n - 1} \). For \( y = u^{-8} \), we have \( \frac{dy}{du} = -8u^{-9} \).

Step3: Differentiate the inner function

For \( u = 7x^2 + x + 7 \), the derivative \( \frac{du}{dx} = 14x + 1 \) (using the power rule for each term: derivative of \( 7x^2 \) is \( 14x \), derivative of \( x \) is \( 1 \), derivative of \( 7 \) is \( 0 \)).

Step4: Apply the chain rule

The chain rule states that \( \frac{dg}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). Substituting the values we found:
\( g'(x) = -8u^{-9} \cdot (14x + 1) \)
Now substitute back \( u = 7x^2 + x + 7 \):
\( g'(x) = -8(7x^2 + x + 7)^{-9}(14x + 1) \)
We can also write this as:
\( g'(x) = \frac{-8(14x + 1)}{(7x^2 + x + 7)^{9}} \)

Answer:

\( -8(14x + 1)(7x^2 + x + 7)^{-9} \) (or equivalently \( \frac{-8(14x + 1)}{(7x^2 + x + 7)^{9}} \))