Questions: Find the derivative of the function.

Transcript text: Find the derivative of the function.

Solution

Solution Steps

To find the derivative of the function \( f(t) = \frac{5}{\sqrt[3]{3t^2 + t}} \), we can use the chain rule and the power rule. First, rewrite the function in a more convenient form for differentiation. Then, apply the chain rule to differentiate the outer function and the inner function separately.

Step 1: Define the Function

We start with the function given by

\[ f(t) = \frac{5}{\sqrt[3]{3t^2 + t}} = 5(3t^2 + t)^{-\frac{1}{3}}. \]

Step 2: Differentiate the Function

To find the derivative \( f'(t) \), we apply the chain rule. The derivative of \( f(t) \) is given by:

\[ f'(t) = 5 \cdot \left(-\frac{1}{3}\right)(3t^2 + t)^{-\frac{4}{3}} \cdot (6t + 1). \]

Step 3: Simplify the Derivative

This simplifies to:

\[ f'(t) = -\frac{5(6t + 1)}{3(3t^2 + t)^{\frac{4}{3}}}. \]