How do you find the derivative of # f(x)=x^3-5x^2+2x+8#?

1 Answer
Jun 30, 2016

Use linearity of differentiation and the power rule to get #f'(x)=3x^2-10x+2#.

Explanation:

The power rule says that #d/dx(x^n)=nx^(n-1)# for any #n#. Therefore, #d/dx(x^3)=3x^2#, #d/dx(x^2)=2x#, #d/dx(x)=1x^(0)=1#, and #d/dx(8)=d/dx(8x^(0))=0#.

Linearity of differentiation says that #d/dx(a_{1}f_{1}(x)+a_{2}f_{2}(x)+\cdots+a_{n}f_{n}(x))#

#=a_{1}f_{1}'(x)+a_{2}f_{2}'(x)+\cdots+a_{n}f_{n}'(x)#, where
#a_{1},a_{2},\ldots,a_{n}# are constants.

These facts lead to

#f'(x)=d/dx(x^3-5x^2+2x+8)#

#=d/dx(x^3)-5d/dx(x^2)+2d/dx(x)+d/dx(8) =3x^2-10x+2#.