**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

^{x.}

^{}

^{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) = 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

Since lne = 1 d/dx[f(x)] = e^{x}^{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) = e3x

^{2}( Note this is not e.3x

^{2}but e with the exponent 3x

^{2})

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)] = e3x

^{2}.(3x^{2})’
= e3x

^{2}(6x)
= 6xe3x

^{2}

**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: 3x

^{2}
1) f(x) = e

^{6x}
2) f(x) = e3x

^{2}-4x+3 ( 3x^{2}-4x+3 is the exponent )
3) f(x) = e

^{x}-e^{-x}/e^{x}-e^{-x}^{}

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