Question

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

Answer 1

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#.