What is the derivative of #f(x) = x^5 * (x^2-3)^#?

1 Answer
Apr 21, 2017

Answer:

#7x^6-15x^4#

Explanation:

Using the product rule:

#d/dx (uw) = u'w + uw'#

This basically means in order to find the derivative of the function #(uw)#, you have to separate it into 2 separate functions, #u# and #w#. Then you find the derivative of #u# and #w# and multiply their derivatives #u'# and #w'# with the other function and add their products together.

Where
#color(white)(aaaaaaa)#(u)#color(white)(aaaaaa)#(w)
#color(white)(aaaaaaa)##darr##color(white)(aaaaa)##darr#
#f(x) = (x^5)*(x^2-3)#

#u = x^5##color(white)(aaaaaa)##w=(x^2-3)#
#u' = 5x^4##color(white)(aaaa)##w'=2x#

#-------------------#

#=(5x^4)(x^2-3) + (2x)(x^5)#
#=5x^6-15x^4+2x^6#
#=7x^6-15x^4#