Ryan Osborne
06/10/2024 · Junior High School
Find the derivative of the function. \[ r(x)=\left(\ln \left(x^{2}\right)\right)^{8} \]
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Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( (\ln(x^2))^8 \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\left(\ln{\left(x^{2}\right)}\right)^{8}\right)\)
- step1: Use differentiation rules:
\(\frac{d}{dg}\left(g^{8}\right)\times \frac{d}{dx}\left(\ln{\left(x^{2}\right)}\right)\)
- step2: Find the derivative:
\(8g^{7}\times \frac{d}{dx}\left(\ln{\left(x^{2}\right)}\right)\)
- step3: Calculate:
\(8g^{7}\times \frac{2}{x}\)
- step4: Substitute back:
\(8\left(\ln{\left(x^{2}\right)}\right)^{7}\times \frac{2}{x}\)
- step5: Multiply the terms:
\(\frac{8\left(\ln{\left(x^{2}\right)}\right)^{7}\times 2}{x}\)
- step6: Multiply the terms:
\(\frac{16\left(\ln{\left(x^{2}\right)}\right)^{7}}{x}\)
- step7: Simplify:
\(\frac{2^{11}\left(\ln{\left(x\right)}\right)^{7}}{x}\)
- step8: Evaluate the power:
\(\frac{2048\left(\ln{\left(x\right)}\right)^{7}}{x}\)
The derivative of the function \( r(x)=\left(\ln \left(x^{2}\right)\right)^{8} \) with respect to \( x \) is \( \frac{2048(\ln{(x)})^{7}}{x} \).
Quick Answer
The derivative is \( \frac{2048(\ln{(x)})^{7}}{x} \).
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