How do you differentiate #f(x)= (x + 7)^10 (x^2 + 2)^7 # using the product rule?

1 Answer
Dec 14, 2015

Product Rule:
If we have two functions #f(x)# and #g(x)#, then
#d/dx# #f(x)g(x)# = #f^'(x)g(x) + f(x)g^'(x)#

Chain Rule:
If we have two functions, #f(x)# and #g(x)#, then
#d/dx# #f(g(x))#=#g^'(x)•f^'(g(x))#

Original Equation:
#f(x)=(x+7)^10(x^2+2)^7#

Use the chain rule to find the derivative of the first equation:
#(x+7)^10#
#(1)•(10)(x+7)^9#
#10(x+7)^9#

Now multiply by the second function:
#10(x+7)^9 (x^2+2)^7#

Next, use the chain rule to find the derivative of the second equation:
#(x^2+2)^7#
#(2x)(7)(x^2+2)^6#
#(14x)(x^2+2)^6#

Multiply by the first function:
#(x+7)^10(14x)(x^2+2)^6#

Add together the two multiplied functions:
#(10(x+7)^9 (x^2+2)^7)#+#((x+7)^10(14x)(x^2+2)^6)#

Factor out #(x+7)^9# and #(x^2+2)^6#:
#{ (x+7)^9(x^2+2)^6} 10(x^2+2)+(x+7)(14x)#

Multiply and combine like terms for the factored section:
#10(x^2+2)+(x+7)(14x)#
#10x^2+20+14x^2+98x#
#24x^2+98x+20#

Put back the factors.
Your final answer should be:
#f^'(x)={ (x+7)^9(x^2+2)^6}(24x^2+98x+20)#