Math / Calculus

QuestionQuestion 9 of 11
Find the derivative of the function. $g(x)=\left(\log _{2}(x)\right)^{9 / 8}$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $g^{\prime}(x)=$

Studdy Solution

STEP 1

1. We are given the function $g(x) = \left(\log_{2}(x)\right)^{9/8}$ .
2. We need to find the derivative $g'(x)$ .

STEP 2

1. Apply the chain rule to differentiate the outer function.
2. Differentiate the inner function, which involves a logarithm with base 2.
3. Combine the results from the chain rule to find the derivative.

STEP 3

Identify the outer function and apply the power rule. The outer function is $u^{9/8}$ where $u = \log_{2}(x)$ .
The derivative of $u^{9/8}$ with respect to $u$ is:
$\frac{9}{8} u^{(9/8) - 1} = \frac{9}{8} u^{1/8}$

STEP 4

Differentiate the inner function $\log_{2}(x)$ . Recall that the derivative of $\log_{a}(x)$ is:
$\frac{1}{x \ln(a)}$
For $\log_{2}(x)$ , the derivative is:
$\frac{1}{x \ln(2)}$

STEP 5

Combine the derivatives using the chain rule. The chain rule states:
$g'(x) = \frac{d}{dx} \left[ u^{9/8} \right] \cdot \frac{d}{dx} \left[ \log_{2}(x) \right]$
Substitute the derivatives from Step 1 and Step 2:
$g'(x) = \frac{9}{8} \left(\log_{2}(x)\right)^{1/8} \cdot \frac{1}{x \ln(2)}$
Simplify the expression:
$g'(x) = \frac{9}{8x \ln(2)} \left(\log_{2}(x)\right)^{1/8}$
The derivative of the function is:
$g'(x) = \frac{9}{8x \ln(2)} \left(\log_{2}(x)\right)^{1/8}$

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