In this problem, we are asked to differentiate the function \( y = (5x + 1)^4 \). To do this, we will apply the **Chain Rule**, which is used when differentiating a composite function (a function within another function).
The function \( (5x + 1)^4 \) is a composite function because it involves an outer function \( u^4 \) where \( u = 5x + 1 \) and an inner function \( 5x + 1 \).
According to the **Chain Rule**, if you have a function \( y = [u(x)]^n \), where \( u(x) \) is a function of \( x \), the derivative of \( y \) with respect to \( x \) is given by:
dy/dx = n [u(x)]^{n-1} * du/dx
Let's identify our functions:
- Outer function: \( [u]^4 \), where \( u = 5x + 1 \)
- Inner function: \( u = 5x + 1 \)
First, differentiate the outer function:
- The derivative of \( [u]^4 \) with respect to \( u \) is \( 4[u]^3 \).
Next, differentiate the inner function:
- The derivative of \( u = 5x + 1 \) with respect to \( x \) is 5.
Now, substitute these values into the chain rule formula:
- dy/dx = 4(5x + 1)^3 * 5
- dy/dx = 20(5x + 1)^3
Therefore, the derivative of \( y = (5x + 1)^4 \) is 20(5x + 1)^3.