## Exponential functions

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

d/(dx)e^x = e^x.

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

d/(dx)b^x = b^x*lnb.

### Example

Let’s differentiate 3x^3(7)^(picos2x).

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

d/(dx)3x^3(7)^(picos2x) = d/(dx)3x^3 * 7^(picos2x) + d/(dx)7^(picos2x) * 3x^3,

which reduces to

9x^2(7)^(picos2x) + 7^(picos2x)*ln7*d/(dx)picos2x,

which then becomes

(9x^2 + ln7d/(dx)picos2x)7^(picos2x),

(9x^2 - ln7*2pisin2x)7^(picos2x).