Log/Exponential Functions Revision Notes for Leaving Cert Mathematics

Log/Exponential Functions

Logarithmic and Exponential Functions

An exponential function is a function in which the variable appears in the exponent. A common example is:

f(x) = 2^x

where the base ( $2$ ) is raised to the power of $x$ .

The natural logarithm function, denoted as $\ln(x)$ , is the inverse of the exponential function with base e.

The key differentiation rules for logarithmic and exponential functions are as follows:

\frac{d}{dx} \ln(x) = \frac{1}{x}

\frac{d}{dx} e^x = e^x

\frac{d}{dx} e^{ax} = a e^{ax}

\frac{d}{dx} a^x = a^x \ln(a)

When differentiating more complex expressions involving exponentials and logarithms, the chain rule is often required.

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For example, if $f(x) = e^{g(x)}$ , then:

f'(x) = e^{g(x)} \times g'(x)

Similarly, for logarithmic functions, the logarithmic differentiation technique can be useful when dealing with products, quotients, or exponentials with variable bases.

Example

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Differentiate $e^{\sin{(x)}}$

Identify an inner and outer function :

v=\sin(x)

u=e^v

Differentiate both functions :

v'= \cos(x)

u'= e^v

Apply chain rule :

\begin{align*} u' \cdot v' &= \cos(x) \cdot e^v \\\\ &= \cos(x)e^{\sin(x)} \end{align*}

Log/Exponential Functions (Leaving Cert Mathematics): Revision Notes