Introduction to derivative:

In calculus the derivative is a measure of how a function changes as its input changes. A derivative can be thought of as how much one quantity is changing in response to changes in some other quantity. Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x.Source Wikipedia.

Derivative formula for exponential functions:

1. `d/dx (e^x)`   = `e^x`

2. `d/dx (e^(ax))` = `a e^(ax)`

3. `d/dx (e^u)` =`e^u (du)/(dx)`

4. `d/dx (b^u)` = `b^u ln b (du)/(dx)`

Exponential derivative problems:

Exponential derivative problem 1:

Find the derivative of given exponential function  3e x

Solution:

` d/dx` (3e x )  = 3 `d/dx` ex

we know the derivative exponential formula `d/dx` `(e^(x))` = ` e^x`

So,                    = 3 `d/dx` e x

= (3) e

= 3 e x

Answer: The derivative of  3 e i s    3 e

Exponential derivative problem 2:

Find the derivative of given exponential function  3sin ex

Solution:

Let   u = `e^(x)`   and   y = 3sin u

`(dy)/(dx)` = `((dy)/(du))` `((du)/(dx))`

So,      `(du)/(dx)``e^(x)`

`(dy)/(du)`   = 3cos u                                      (derivative of sin u = cos u)

`(dy)/(dx)` = `((dy)/(du))`` ((du)/(dx))`

= 3 cos u ` [e^(x) ]`

= 3 cos ex (ex)

= 3 ex  cos ex

Answer: The derivative of  3 sin ex  is    3 ex  cos ex

Exponential derivative problem 3:

Find the derivative of given exponential function  `cos e^(3x) `

Solution:

Let   u = `e^(3x)`   and   y = cos u

`(dy)/(dx)` = `((dy)/(du))` `((du)/(dx))`

So,      `(du)/(dx)` = 3 `e^(3x)`

`(dy)/(du)`   = -sin u

derivative of sin u = cos u)

`(dy)/(dx)` = `((dy)/(du))`` ((du)/(dx))`

= - sin u ` [3 e^(3x) ]`

= `- sin e^(3x) [3 e^(3x) ]`

= ` - 3 e^(3x)`   `sine^(3x)`

Answer:  The derivative of  cos e3x  is  ` - 3 e^(3x)`   `sine^(3x)`

Exponential derivative problem 4:

Find the derivative of given exponential function 4e3x

Solution:

` d/dx` (4e3x)  = 4 `d/dx` e3x

we know the derivative exponential formula `d/dx` `(e^(ax))` = `a e^ax`

Here, a = 3

So,                         = 4 `d/dx` e3x

= (4) e3x

= 12 e3x

Answer: The derivative of  4 e3x is 12 e3x