Questions: Find the derivative of the function. f(v)=(v^-3+5 v^-2)^3 f'(v)=□

Transcript text: Find the derivative of the function. \[ \begin{array}{l} f(v)=\left(v^{-3}+5 v^{-2}\right)^{3} \\ f^{\prime}(v)=\square \end{array} \]

Solution

Solution Steps

Step 1: Define the Function

Given the function: \[ f(v) = \left( v^{-3} + 5v^{-2} \right)^3 \]

Step 2: Apply the Chain Rule

To find the derivative \( f'(v) \), we use the chain rule. First, differentiate the outer function, keeping the inner function unchanged: \[ \frac{d}{dv} \left[ \left( v^{-3} + 5v^{-2} \right)^3 \right] = 3 \left( v^{-3} + 5v^{-2} \right)^2 \cdot \frac{d}{dv} \left( v^{-3} + 5v^{-2} \right) \]

Step 3: Differentiate the Inner Function

Next, we differentiate the inner function: \[ \frac{d}{dv} \left( v^{-3} + 5v^{-2} \right) = -3v^{-4} - 10v^{-3} \]

Step 4: Combine the Results

Now, we combine the results from the chain rule: \[ f'(v) = 3 \left( v^{-3} + 5v^{-2} \right)^2 \cdot \left( -3v^{-4} - 10v^{-3} \right) \]

Step 5: Simplify the Expression

Simplify the resulting expression: \[ f'(v) = 3 \left( v^{-3} + 5v^{-2} \right)^2 \cdot \left( -3v^{-4} - 10v^{-3} \right) \] \[ f'(v) = \left( -9v^{-4} - 30v^{-3} \right) \left( v^{-3} + 5v^{-2} \right)^2 \]

Step 6: Further Simplification

Further simplifying the expression, we get: \[ f'(v) = \frac{(-30v - 9)(5v + 1)^2}{v^{10}} \]