Questions: Let f(x)=(-9x^2-5)^5(4x^2-6)^9. Find the derivative and write answer in factored form.

Transcript text: Let $f(x)=\left(-9 x^{2}-5\right)^{5}\left(4 x^{2}-6\right)^{9}$. Find the derivative and write answer in factored form.

Solution

Solution Steps

To find the derivative of the function $ f(x) = \left(-9 x^{2}-5\right)^{5}\left(4 x^{2}-6\right)^{9} $, we will use the product rule and the chain rule. The product rule states that the derivative of a product of two functions is given by $ (uv)' = u'v + uv' $. The chain rule will be used to differentiate each of the composite functions.

Step 1: Define the Function

We start with the function defined as: \[ f(x) = (-9x^2 - 5)^5 (4x^2 - 6)^9 \]

Step 2: Compute the Derivative

Using the product rule and chain rule, we find the derivative $ f'(x) $: \[ f'(x) = 72x(-9x^2 - 5)^5(4x^2 - 6)^8 - 90x(-9x^2 - 5)^4(4x^2 - 6)^9 \]

Step 3: Factor the Derivative

The derivative can be factored as follows: \[ f'(x) = -9216x(2x^2 - 3)^8(9x^2 + 5)^4(28x^2 - 5) \]