Exponential functions

The derivative of the exponential function is, by definition, itself:

$\frac{d}{dx}{e}^x=e^x\text{.}$

With other bases, you have to multiply by the natural logarithm of the base:

$\frac{d}{dx}{b}^x=b^x\cdot \ln b\text{.}$

Example

Let’s differentiate $3x^3(7{)}^{\pi \cos 2x}\text{.}$

The first thing to notice is that we can use the product rule:

$\frac{d}{dx}3x^3(7{)}^{\pi \cos 2x}=\frac{d}{dx}3x^3\cdot 7^{\pi \cos 2x}+\frac{d}{dx}{7}^{\pi \cos 2x}\cdot 3x^3\text{,}$

which reduces to

$9x^2(7{)}^{\pi \cos 2x}+7^{\pi \cos 2x}\cdot \ln 7\cdot \frac{d}{dx}\pi \cos 2x\text{,}$

which then becomes

$(9x^2+\ln 7\frac{d}{dx}\pi \cos 2x)7^{\pi \cos 2x}\text{,}$

giving us the final answer,

$(9x^2-\ln 7\cdot 2\pi \sin 2x)7^{\pi \cos 2x}\text{.}$