How do you differentiate f(x) = 3x(e^((x-9)^2))^3f(x)=3x(e(x9)2)3 using the chain rule?

1 Answer
Jun 1, 2017

e^((x-9)^6)(18x(x-9)^5+3)e(x9)6(18x(x9)5+3)

Explanation:

Step 1. Simplify the exponent

f(x)=3x e^((x-9)^6)f(x)=3xe(x9)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)