Questions: Find f'(x): 10. f(x) = (4x^2)^3 11. f(x) = ln(x)

Find f'(x): 10. f(x) = (4x^2)^3 11. f(x) = ln(x)
Transcript text: Find $f^{\prime}(x)$ : 10. $f(x)=\left(4 x^{2}\right)^{3}$ 11. $f(x)=\ln (x)$
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Solution

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Solution Steps

To find the derivatives of the given functions, we will use the rules of differentiation. For the first function, we will apply the chain rule and power rule. For the second function, we will use the derivative of the natural logarithm function.

Step 1: Differentiate \( f(x) = (4x^2)^3 \)

To find the derivative of the function \( f(x) = (4x^2)^3 \), we first simplify it: \[ f(x) = 64x^6 \] Now, we differentiate: \[ f'(x) = \frac{d}{dx}(64x^6) = 384x^5 \]

Step 2: Differentiate \( f(x) = \ln(x) \)

Next, we differentiate the function \( f(x) = \ln(x) \): \[ f'(x) = \frac{d}{dx}(\ln(x)) = \frac{1}{x} \]

Final Answer

The derivatives are:

  1. For \( f(x) = (4x^2)^3 \), \( f'(x) = 384x^5 \)
  2. For \( f(x) = \ln(x) \), \( f'(x) = \frac{1}{x} \)

Thus, the final answers are: \[ \boxed{f'(x) = 384x^5} \] \[ \boxed{f'(x) = \frac{1}{x}} \]

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