(3x+1)^3/(3x+1)^3 +1 use the chain rule to complete the derivative

if u = 3x+1, we have

y = u^3/(u^3+1)
y' = 3u^2/(u^3+1)^2 u'
= 9(3x+1)^2/((3x+1)^3 + 1)^2

Thank you

To find the derivative of the given expression using the chain rule, follow these steps:

Step 1: Identify the outer function and the inner function. In this case, the outer function is (3x+1)^3, and the inner function is 3x+1.

Step 2: Find the derivative of the outer function. In this case, the derivative of (3x+1)^3 is 3(3x+1)^2 multiplied by the derivative of the inner function.

Step 3: Find the derivative of the inner function, which is 3x+1. Since the derivative of x with respect to x is 1, and the derivative of a constant (1 in this case) is 0, the derivative of 3x+1 is simply 3.

Step 4: Multiply the derivative of the outer function by the derivative of the inner function to obtain the overall derivative.

Using the chain rule, the derivative of the given expression is:

3(3x+1)^2 * 3

Simplifying further:

9(3x+1)^2

Therefore, the derivative of (3x+1)^3/(3x+1)^3 + 1 using the chain rule is 9(3x+1)^2.