# What is the antiderivative of 4x? thanks!?

Jun 8, 2015

Normally you would use the Power Rule like so to do the derivative:

$\frac{d}{\mathrm{dx}} \left[k {x}^{n}\right] = n \cdot k {x}^{n - 1}$
where $k$ is a coefficient and $n$ is a real number.

With the antiderivative of something on which you can use the power rule to take the derivative, you can do this backwards.

$F \left(k {x}^{n}\right) = \frac{k {x}^{n + 1}}{n + 1}$

So:

$F \left(4 x\right) = \frac{4 {x}^{1 + 1}}{1 + 1} = \frac{4 {x}^{2}}{2} = 2 {x}^{2}$

Notice how $\frac{d}{\mathrm{dx}} \left[2 {x}^{2}\right] = 2 \cdot 2 {x}^{2 - 1} = 4 x$, so it worked.