Saturday, June 10, 2017

Derivative of exponential functions

Derivative of f(x) = b

In the expression above b is a positive real number and is called the base of the exponential function.
The formula to calculate the derivative is d/dx[f(x)] = lnb.bx.

Rule: The derivative of an exponential function is equal to the product of the natural logarithm of the base by the function.

Example 1 calculate the derivative of f(x) = 2x

The given function has the form f(x) = bx. By applying the formula d/dx[f(x)] = lnb. bx  d/dx[f(x)] = ln2.2x

Derivative of f(x) = bu 


Since f is a composite function where u is a function of x the derivative of f is d/dx [f(x)] = d/du(bu).du/dx= lnb.bu .u'


Rule: The derivative of an exponential function with base b is equal to the product of the natural logarithm of the base by the derivative of u.


Example 2. Calculate the derivative of f(x) = 32x

Let’s apply the formula for the derivative of f(x) = bu which is d/dx[f(x)] = lnb.bu.u’
d/dx[f(x)] = ln3.32x(2x)’
                = ln3.32x.2
                = 2ln3.32x
Derivative of f(x) = ex

The derivative f(x) = eis a special case of f(x) = bx where b = e

Let's substitute b in the formula d/dx[f(x)] = lnb.bx
d/dx[f(x)] = lne.ex
Since lne = 1 d/dx[f(x)] = ex

Rule: The derivative of the function f(x) = eis the function eitself. 


Derivative of f(x) = eu 


Since f is a composite function where u is a function of x its derivative is given by the derivative of a composite function.

Then d/dx[f(x)] = d/du(eu).du/dx = eu.u’

Rule: The derivative of the composite exponential function with base e is equal to the product of the composite function by the derivative of the function u.



Example 3. Calculate the derivative of f(x) = e3x2 ( Note this is not e.3x2 but e with the exponent 3x2)
Let’s apply the formula for the derivative of f(x) = eu which is d/dx[f(x)] = eu.u’
d/dx[f(x)] = e3x2.(3x2)’ 
                = e3x2(6x)
                = 6xe3x2


Summary


The derivative of f(x) = bwhere b>0 is d/dx(bx) - lnb.bx

The derivative of the composite function f(x) = bu where u is a function of x is d/dx(bu) = lnb.bu u’
The derivative of f(x) = ex is d/dx(ex) = ex
 The derivative of f(x) = eis d/dx(eu) = eu.u

Practice
Calculate the derivative of the following functions: 3x2
1) f(x) = e6x 
2) f(x) = e3x2-4x+3 ( 3x2-4x+3 is the exponent )
3) f(x) = ex-e-x/ex-e-x

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