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

Nov 23, 2015

$f ' \left(x\right) = 3 {x}^{2} - 32 x + 30$

#### 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, $f ' \left(x\right) = \stackrel{\text{chain rule"overbrace(d/dx[(x-5)^3])-stackrel"nx^(n-1)"overbrace(d/dx[x^2])+stackrel"nx^(n-1)}}{\overbrace{\frac{d}{\mathrm{dx}} \left[5 x\right]}}$

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

Through the Chain Rule:

$\frac{d}{\mathrm{dx}} \left[{\left(x - 5\right)}^{3}\right] = 3 {\left(x - 5\right)}^{2} \cdot \frac{d}{\mathrm{dx}} \left[x\right] = 3 {\left(x - 5\right)}^{2} \cdot 1 = 3 {\left(x - 5\right)}^{2}$

$\frac{d}{\mathrm{dx}} \left[{x}^{2}\right] = 2 x$

$\frac{d}{\mathrm{dx}} \left[5 x\right] = 5$

We can add all these back together:

$f ' \left(x\right) = 3 {\left(x - 5\right)}^{2} - 2 x + 5$

And, simplify:

$f ' \left(x\right) = 3 \left({x}^{2} - 10 x + 25\right) - 2 x + 5$

$f ' \left(x\right) = 3 {x}^{2} - 30 x + 25 - 2 x + 5$

$f ' \left(x\right) = 3 {x}^{2} - 32 x + 30$