How do you differentiate #f(x) = 3x(e^((x-9)^2))^3# using the chain rule?

1 Answer
Jun 1, 2017

Answer:

#e^((x-9)^6)(18x(x-9)^5+3)#

Explanation:

Step 1. Simplify the exponent

#f(x)=3x e^((x-9)^6)#

Step 2. Using the Product Rule directly uses the chain rule.

#f'(x)=3x d/dx(e^((x-9)^6))+e^((x-9)^6) d/dx(3x)#

#=3x e^((x-9)^6) d/dx((x-9)^6)+3e^((x-9)^6)#

#=3x e^((x-9)^6)(6(x-9)^5)+3e^((x-9)^6)#

#=18x(x-9)^5 e^((x-9)^6)+3e^((x-9)^6)#

Step 3. Factor out the common #e^((x-9)^6)# term

#=e^((x-9)^6)(18x(x-9)^5+3)#