Exponential functions can be differentiated using the chain rule. One of the most intriguing and functional characteristics of the natural exponential function is that *it is its own derivative*.

In other words, it has solution to the differential equation being the same such that,*y’* = *y*.The exponential function which has the property that the slope of the tangent line at (0,1) has the value *m*_{0} = 1 is called the **natural exponential function**. It can be written as exp(*x*) or more frequently as *e ^{x}*, where

*e*is the base of the exponential function which has a unit slope at (0,1).

Let u be a differentiable function of x, then we can accept the following properties:

This can be proved using the limit formula:

1 | 1.71828 |

0.1 | 1.05171 |

0.01 | 1.00502 |

0.001 | 1.00050 |

0.0001 | 1.00005 |

0.00001 | 1.00001 |

We can use the chain rule to find the derivative of the a^{x}.

Example: Find the derivative of

Example: Differentiate the following function: y = e^{tan x}

Solution: Use chain rule, let u = tan x

Then, we have y = e^{u}.

Therefore,

A derivative of natural logarithm function:

Let such that x â 0

Then,

Example: Differentiate*y* = ln(*x*^{3} + 1).

Solution, Use chain rule to solve this function:

Let *u *=* x*^{3}+1, then y = ln u

Example: Find

Let *y* = log* _{a}x. *Then,

*a*=

^{y}*x*.

We can differentiate the equation above implicitly with respect to *x*, we obtain:

Therefore, we obtain:

If we put *a *=* e *in the above formula, then the factor on the right side becomes ln* e *=1and we get the formula for the derivative of the natural logarithmic function log* _{e }x *=ln

*x.*

The differentiation formula can be manipulated in the simplest form when *a *=* e *because ofln* e *=1.

Example: Find *f â*(*x*) if *f*(*x*) = ln |*x*|.

Solution: We know:

It follows that

Thus, fâ(x) = 1/x for all x not equal to 0.