What is the derivative of #g(x)=x^3 cos x#?

1 Answer
Mar 25, 2018

Answer:

#g'(x)= 3x^2cosx-x^3sinx#

Explanation:

Since #g(x)# is the product of two terms, we can use the Product Rule to find the derivative.

We essentially have #g(x)=f(x)*h(x)#, where

#color(purple)(f(x)=x^3)# and

#color(green)(h(x)=cosx)#

Thus, the Product Rule states that the derivative is equal to:

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

To differentiate #f(x)#, we can use the Power Rule, where the exponent becomes the coefficient, and we decrement the power. Thus,

#color(blue)(f'(x)=3x^2)#

And from our knowledge of derivatives of trig functions

#color(red)(h'(x)=-sinx)#

We can now plug these values into the product rule expression to get

#color(blue)(3x^2)color(green)((cosx))+color(purple)(x^3)color(red)((-sinx))#

We can rewrite this as

#3x^2(cosx)-x^3(sinx)#

Thus, #g'(x)= 3x^2cosx-x^3sinx#

If the Power or Product Rules seem foreign to you, I encourage you to Google them or go to Khan Academy to understand them more.

Hope this helps!