How do you differentiate #f(x)=(1/x^3)sinx# using the product rule?

1 Answer
May 25, 2018

Answer:

#-3x^-4sinx+x^-3cosx#

Explanation:

Let's start by rewriting #f(x)# as #x^(-3)sinx#. This allows us to differentiate it more easily.

We understand that if we have two functions, #g(x)# and #h(x)# being multiplied, the derivative is equal to

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

#f(x)# is essentially composed of the following:

  • #color(blue)(g(x)=x^-3)#
  • #color(red)(h(x)=sinx)#

We can use the power rule and our knowledge of trig derivatives to figure out that

  • #color(purple)(g'(x)=-3x^-4)#
  • #color(lime)(h'(x)=cosx)#

Now, we just plug this business into our expression for the product rule. We get

#color(purple)(-3x^-4)color(red)((sinx))+color(blue)(x^-3)color(lime)((cosx))#

And I can write this with neutral colors to get

#bar(ul(|color(white)(2/2)-3x^-4sinx+x^-3cosxcolor(white)(2/2)|))#

Hope this helps!