Question

How do you differentiate `f(x)=(x-5)^3-x^2+5x` using the sum rule?

Answer 1

`f'(x)=3x^2-32x+30`

Explanation:

The sum rule is simple. All we have to do is find the derivative of each part of the sum and add them back to one another.

Therefore, `f'(x)=stackrel"chain rule"overbrace(d/dx[(x-5)^3])-stackrel"nx^(n-1)"overbrace(d/dx[x^2])+stackrel"nx^(n-1)"overbrace(d/dx[5x])`

I've written the rules we'll need to continue in finding the derivatives.

Through the Chain Rule:

`d/dx[(x-5)^3]=3(x-5)^2*d/dx[x]=3(x-5)^2*1=3(x-5)^2`

`d/dx[x^2]=2x`

`d/dx[5x]=5`

We can add all these back together:

`f'(x)=3(x-5)^2-2x+5`

And, simplify:

`f'(x)=3(x^2-10x+25)-2x+5`

`f'(x)=3x^2-30x+25-2x+5`

`f'(x)=3x^2-32x+30`