Derivative of #x^5(3-6x^-9)#?

2 Answers
Feb 9, 2018

Answer:

#f'(x)=54x^-5+5x^4(3-6x^-9)#

Explanation:

We have #f(x)=x^5(3-6x^-9)=g(x)h(x)#

#f'(x)=g(x)h'(x)+g'(x)h(x)#

#g(x)=x^5#
#g'(x)=5x^4#

#h(x)=3-6x^-9#
#h'(x)=-9*-6x^(-9-1)=54x^-10#

#f'(x)=x^5(54x^-10)+5x^4(3-6x^-9)#
#color(white)(f'(x))=54x^-5+5x^4(3-6x^-9)#

Feb 9, 2018

Answer:

#f^'(x)=15x^4+24x^(-5)#

Explanation:

Set #y=x^5(3-6x^(-9))#

Multiply out the brackets

#y=3x^5-(6x^5)/x^9#

#y=3x^5-6/x^4#

#y=3x^5-6x^(-4)#

Or if you prefer

#f(x)=3x^5-6x^(-4)#

The next step is to differentiate each term.

Using the principle that if: #g(x)=ax^n -> g^'(x) =(axxn)x^(n-1)#

#f^'(x)=(3xx5)x^(5-1)-[6xx(-4)x^(-4-1)]#

#f^'(x)=15x^4+24x^(-5)#

They used to use #dy/dx# for #f^'(x)# You will still see this sometimes.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Check")#
Output from Maple:
Tony B

#5x^4(3-6/x^9)+54/x^5#

#15x^4-30/x^5+54/x^5#

#15x^4-24/x^5#

#15x^4-24x^(-5) color(red)(larr" Confirmed")#