Differentiating e^x and its friends

Remember:

1. e is a number whose value is about 2.72.

2. e is used as the base for the exponential and logarithm functions, e^x and $\ln x$.

The function e^x has the following amazing property:

$$\boxed{\frac{d}{dx} (e^x) = e^x}$$

The derivative of e^x is just e^x again. That's kind of like giving birth to yourself, a very difficult event to achieve, and one that would get you major tabloid coverage.

NEWS FLASH: ``I AM MY OWN MOTHER," FUNCTION CLAIMS!

In fact, e^x and its multiples are the only functions that are their own derivatives. For this reason we put up with the use of the very strange number 2.718281828459... as the base.

Example   Find $$\frac{d}{dx} ( e^ {\sin x}).$$

Notice that $$e^ {\sin x}$$ is the composition of e^x with $\sin x$ . So we'll use the chain rule to differentiate it.

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

The $$e^ {\sin x}$$ is the derivative of the outside function e^uevaluated at the inside function $u = \sin x$ and the $\cos x$ is the derivative of the inside function.