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`.


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)`,

giving us the final answer,

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