# How do you find the fourth derivative of e^(2x)?

Nov 29, 2016

You can use the chain rule to find the first derivative of ${e}^{2 x}$ and differentiate iteratively

#### Explanation:

$\frac{{\mathrm{de}}^{2 x}}{\mathrm{dx}} = \frac{{\mathrm{de}}^{2 x}}{d \left(2 x\right)} \cdot \frac{d \left(2 x\right)}{\mathrm{dx}} = 2 {e}^{2 x}$

$\frac{{d}^{\left(2\right)} {e}^{2 x}}{\mathrm{dx}} ^ 2 = 4 {e}^{2 x}$

$\frac{{d}^{\left(3\right)} {e}^{2 x}}{\mathrm{dx}} ^ 3 = 8 {e}^{2 x}$

$\frac{{d}^{\left(4\right)} {e}^{2 x}}{\mathrm{dx}} ^ 4 = 16 {e}^{2 x}$

In general it is easy to see that:

$\frac{{d}^{\left(n\right)} {e}^{\alpha x}}{\mathrm{dx}} ^ n = {\alpha}^{n} {e}^{\alpha x}$