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

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Answer 1 · 2015-11-23T01:15:26

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

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:

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

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

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

How do you differentiate f(x)=(x-5)^3-x^2+5xf(x)=(x−5)3−x2+5xf(x)=(x-5)^3-x^2+5x using the sum rule?