Exponential functions

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

ddxex=ex.

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

ddxbx=bxlnb.

Example

Let’s differentiate 3x3(7)πcos2x.

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

ddx3x3(7)πcos2x=ddx3x37πcos2x+ddx7πcos2x3x3,

which reduces to

9x2(7)πcos2x+7πcos2xln7ddxπcos2x,

which then becomes

(9x2+ln7ddxπcos2x)7πcos2x,

giving us the final answer,

(9x2ln72πsin2x)7πcos2x.