How do you differentiate #f(x)=((65e^-7x)+2)^3 # using the chain rule?

Question

Elijah Abram · Accepted Answer

Take the derivative of the outside, and leave the inside alone, then multiply it by the derivative of the inside.

#f(x)=((65e^(-7)x+2)^3# Outermost layer is the #( )^3# So... #f'(x)=3(65e^(-7)x+2)^2# is the first piece. Then multiply that times the derivative of the inside, which, despite the confusing look of the first term, is a simple linear equation... so... #f'(x)=3(65e^(-7)x+2)^2*(65e^(-7))#

Evelyn Agard · Answer

undefined To differentiate $ f(x) = ((65e^{-7x}) + 2)^3 $ using the chain rule, follow these steps:

1. Identify the outer function and the inner function.
2. Find the derivative of the outer function with respect to its inner function.
3. Find the derivative of the inner function with respect to $ x $.
4. Multiply the results of steps 2 and 3 together to get the final derivative.

Let's break it down:

1. Outer function: $ g(u) = u^3 $, where $ u = 65e^{-7x} + 2 $.
2. Derivative of the outer function: $ g'(u) = 3u^2 $.
3. Inner function: $ u = 65e^{-7x} + 2 $.
4. Derivative of the inner function: $ u' = \frac{d}{dx}(65e^{-7x} + 2) = -455e^{-7x} $.
5. Multiply the derivatives: $ g'(u) \cdot u' = 3(65e^{-7x} + 2)^2 \cdot (-455e^{-7x}) $.
6. Simplify the expression: $ \frac{d}{dx}((65e^{-7x} + 2)^3) = -1365(65e^{-7x} + 2)^2e^{-7x} $.

So, the derivative of $ f(x) = ((65e^{-7x}) + 2)^3 $ using the chain rule is $ -1365(65e^{-7x} + 2)^2e^{-7x} $.