Use the chain rule to find the derivative of the following function.𝑓(𝑟)=𝑒6𝑟2+2𝑟+2

Sure, here is the step-by-step solution:

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Given the function f(r) = e^(6r^2 + 2r + 2), we can identify the outer function as e^x and the inner function as 6r^2 + 2r + 2.

Step 1: Find the derivative of the outer function. The derivative of e^x with respect to x is e^x.

Step 2: Find the derivative of the inner function. The derivative of 6r^2 + 2r + 2 with respect to r is 12r + 2.

Step 3: Apply the chain rule. According to the chain rule, the derivative of f(r) is the derivative of the outer function times the derivative of the inner function. Therefore, the derivative of f(r) is e^(6r^2 + 2r + 2) * (12r + 2).

So, f'(r) = (12r + 2)e^(6r^2 + 2r + 2).