Questions: Find the derivative of the following function. y=ln (x^2+9)^π dy/dx= (Type an exact answer, using π as needed.)

Transcript text: Find the derivative of the following function. \[ y=\ln \left(x^{2}+9\right)^{\pi} \] \[ \frac{d y}{d x}= \] (Type an exact answer, using $\pi$ as needed.)

Solution

Solution Steps

Step 1: Differentiate the Function

To find the derivative of the function $ y = \ln((x^2 + 9)^{\pi}) $, we apply the chain rule. The derivative of the natural logarithm function is given by:

\[ \frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx} \]

where $ u = (x^2 + 9)^{\pi} $.

Step 2: Differentiate the Inner Function

Next, we differentiate the inner function $ u $. Using the power rule, we have:

\[ \frac{du}{dx} = \pi (x^2 + 9)^{\pi - 1} \cdot \frac{d}{dx}(x^2 + 9) = \pi (x^2 + 9)^{\pi - 1} \cdot 2x \]

Step 3: Combine the Results

Now, substituting back into the derivative of the logarithm, we get:

\[ \frac{dy}{dx} = \frac{1}{(x^2 + 9)^{\pi}} \cdot \left( \pi (x^2 + 9)^{\pi - 1} \cdot 2x \right) \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{2\pi x}{x^2 + 9} \]