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A derivative is simply a rate of change. Whether you're modeling the movement of a particle or a supply/demand model, this is a key instrument of Calculus.

When \( f(x) = \ln ( x^{13} ) \), what is the value of \( f'(37)? \)

**Details and assumptions**

\( \ln \) is the logarithm base \( e \), also called the natural logarithm.

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If \( f(x) = 4 \ln (e^ {5} x) \), what is the value of \( f'(1) \)?

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If \( f(x) = \log_{2}{x} \), what is \( f'(x) \)?

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For \(y=6\ln x+9x,\) what is the value of \(x \cdot \frac{dy}{dx} ?\)

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If \( f(x) = \log_{3}{(x+2)} \), what is \( f'(x) \)?

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