Exponential functions

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

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

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

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

Example

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

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

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

which reduces to

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

which then becomes

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

giving us the final answer,

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