Math  /  Calculus

QuestionQuestion 9 of 11
Find the derivative of the function. g(x)=(log2(x))9/8g(x)=\left(\log _{2}(x)\right)^{9 / 8} (Express numbers in exact form. Use symbolic notation and fractions where needed.) g(x)=g^{\prime}(x)=

Studdy Solution
Combine the derivatives using the chain rule. The chain rule states:
g(x)=ddx[u9/8]ddx[log2(x)] 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)=98(log2(x))1/81xln(2) g'(x) = \frac{9}{8} \left(\log_{2}(x)\right)^{1/8} \cdot \frac{1}{x \ln(2)}
Simplify the expression:
g(x)=98xln(2)(log2(x))1/8 g'(x) = \frac{9}{8x \ln(2)} \left(\log_{2}(x)\right)^{1/8}
The derivative of the function is:
g(x)=98xln(2)(log2(x))1/8 g'(x) = \frac{9}{8x \ln(2)} \left(\log_{2}(x)\right)^{1/8}

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