Questions: Evaluate the derivative of the following function. f(t) = ln(arctan(5t))

Solution Steps

To find the derivative of the function \( f(t) = \ln(\tan^{-1}(5t)) \), we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is the natural logarithm, and the inner function is the inverse tangent function.

1. Differentiate the outer function \( \ln(u) \) with respect to \( u \), which gives \( \frac{1}{u} \).
2. Differentiate the inner function \( \tan^{-1}(5t) \) with respect to \( t \), which involves using the chain rule again for \( 5t \).

We start with the function given: \[ f(t) = \ln(\tan^{-1}(5t)) \]

To find the derivative \( f'(t) \), we use the chain rule. The chain rule states: \[ \frac{d}{dt} \ln(u) = \frac{1}{u} \cdot \frac{du}{dt} \] where \( u = \tan^{-1}(5t) \).

Next, we differentiate the inner function \( \tan^{-1}(5t) \) with respect to \( t \): \[ \frac{d}{dt} \tan^{-1}(5t) = \frac{1}{1 + (5t)^2} \cdot 5 = \frac{5}{25t^2 + 1} \]

Now, we combine the results from the chain rule and the inner function differentiation: \[ f'(t) = \frac{1}{\tan^{-1}(5t)} \cdot \frac{5}{25t^2 + 1} \]

Final Answer