Derivative of exponential functions

Derivative of f(x) = b^x 

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.b<sup>x.

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

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

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

Derivative of f(x) = b^u 

Since f is a composite function where u is a function of x the derivative of f is d/dx [f(x)] = d/du(b^u).du/dx= lnb.b^u .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) = 3^2x. 

Let’s apply the formula for the derivative of f(x) = b^u which is d/dx[f(x)] = lnb.b^u.u’

d/dx[f(x)] = ln3.3^2x(2^x)’

                = ln3.3^2x.2

                = 2ln3.3^2x

Derivative of f(x) = e^x

The derivative f(x) = e^x is a special case of f(x) = b^x where b = e
Let's substitute b in the formula d/dx[f(x)] = lnb.b^x

d/dx[f(x)] = lne.e^x

Since lne = 1 d/dx[f(x)] = e^x

Rule: The derivative of the function f(x) = e^x is the function e^x itself. 

Derivative of f(x) = e^u 

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(e^u).du/dx = e^u.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) = e^u which is d/dx[f(x)] = e^u.u’

d/dx[f(x)] = e3x².(3x²)’ 

                = e3x²(6x)

                = 6xe3x²

Summary

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

The derivative of the composite function f(x) = b^u where u is a function of x is d/dx(b^u) = lnb.b^u u’

The derivative of f(x) = e^x is d/dx(ex) = e^x

 The derivative of f(x) = e^u is d/dx(e^u) = e^u.u’

Practice

Calculate the derivative of the following functions: 3x²

1) f(x) = e^6x 

2) f(x) = e3x²-4x+3 ( 3x²-4x+3 is the exponent )

3) f(x) = e^x-e^-x/e^x-e^-x

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