Questions: Differentiate the function. y=(5x-5)^4(5-x^4)^4 dy/dx=

Transcript text: Differentiate the function. \[ \begin{array}{l} y=(5 x-5)^{4}\left(5-x^{4}\right)^{4} \\ \frac{d y}{d x}=\square \end{array} \] $\square$

Solution

Solution Steps

To differentiate the given function, we will use the product rule and the chain rule. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. The chain rule is used to differentiate composite functions. We will apply these rules to find the derivative of the given function.

Step 1: Define the Function

We start with the function given by: \[ y = (5x - 5)^{4}(5 - x^{4})^{4} \]

Step 2: Differentiate the Function

Using the product rule and chain rule, we differentiate $y$ with respect to $x$: \[ \frac{dy}{dx} = -16x^{3}(5 - x^{4})^{3}(5x - 5)^{4} + 20(5 - x^{4})^{4}(5x - 5)^{3} \]

Step 3: Simplify the Derivative

The derivative can be expressed as: \[ \frac{dy}{dx} = (5 - x^{4})^{3}(5x - 5)^{3} \left(-16x^{3}(5 - x^{4}) + 20(5x - 5)\right) \]