If #f(x)= 2 x^2 - 3 x # and #g(x) = 2e^x + 1 #, how do you differentiate #f(g(x)) # using the chain rule?

Question

If #f(x)=  2  x^2  - 3   x # and #g(x)  = 2e^x +  1    #, how do you differentiate #f(g(x))  # using the chain rule?

Isabella Ackerman · Accepted Answer

Hi there!

It looks like you're being asked to differentiate a composite function! You'll want to start off by subbing g(x) into f, then you can differentiate.

Starting off with composing the function, or subbing g(x) into f. Take g(x) and sub it into wherever there is an x: #f(g(x)) = 2(2e^x+1)^2-3(2e^x+1) # Now you can differentiate using the chain rule. Let's differentiate: Remember that when differentiating something of the form: # f(x) = a(g(x))^n#, by the chain rule you get: #f'(x) = an(g(x))^(n-1) * g'(x) # Now, using the same principle you can differentiate: #f(g(x))' = 4(2e^x+1)^1*2e^x -3(2e^x+1)^0*2e^x # Simplifying we get: #f(g(x))' = 4(2e^x+1)*2e^x-3(1)*2e^x # And that's pretty much it! Most teachers/professors are okay with leaving it unsimplified, as cleaning it up doesn't change the answer. Hopefully you've understood how the chain rule was applied! If you have any questions, let me know!

Charles Arocha · Answer

Using a less formal notation on part only as a demonstration

Consider : # 2(2e^x+1)^2# Let #u=2e^x+1 # then #(du)/(dx) = 2e^x# Let # y=2u^2 # then #(dy)/(du) = 4u# But #(dy)/(dx) = (dy)/(du)xx(du)/(dx)# so #(dy)/(dx) = 4uxx2e^x" "=" "4(2e^x+1)xx2e^x# so #(dy)/(dx) = 8e^x(2e^x+1)# Which can be worked on further This is only a demonstration of method

Elijah Abram · Answer

undefined To differentiate $ f(g(x)) $ using the chain rule, follow these steps:

1. Identify the inner function, $ g(x) $, and the outer function, $ f(x) $.
2. Find the derivative of the outer function with respect to its input variable.
3. Find the derivative of the inner function with respect to its input variable.
4. Multiply the derivatives found in steps 2 and 3 together.
5. Substitute the inner function $ g(x) $ and its derivative into the result from step 4.
6. Simplify the expression if possible.

Given $ f(x) = 2x^2 - 3x $ and $ g(x) = 2e^x + 1 $:

1. Inner function: $ g(x) = 2e^x + 1 $
   Outer function: $ f(x) = 2x^2 - 3x $

2. Derivative of the outer function: $ f'(x) = 4x - 3 $

3. Derivative of the inner function: $ g'(x) = 2e^x $

4. Multiply the derivatives: $ f'(g(x)) \cdot g'(x) = (4g(x) - 3) \cdot g'(x) $

5. Substitute $ g(x) = 2e^x + 1 $ and $ g'(x) = 2e^x $:
   $ (4(2e^x + 1) - 3) \cdot 2e^x $

6. Simplify the expression if possible.

This is the process to differentiate $ f(g(x)) $ using the chain rule.