Differentiation of Exponential and Logarithmic Functions

Expert reviewed • 05 March 2025 • 8 minute read

HSC Maths Advanced Syllabus

apply the product, quotient and chain rules to differentiate functions of the form $f(x)g(x),\frac{f(x)}{g(x)}$ $f (x) g (x), \frac{f ( x )}{g ( x )}$ and $f(g(x))$ $f (g (x))$ where $f(x)$ $f (x)$ and $g(x)$ $g (x)$ are any of the functions covered in the scope of this syllabus, for example $xe^x$ $x e^{x}$ , $tanx$ $t an x$ , $\frac{1}{x^n}$ $\frac{1}{x ^{n}}$ , $xsinx$ $x s in x$ , $e^{-x}sinx$ $e^{- x} s in x$ and $f(ax+b)$ $f (a x + b)$
- use the composite function rule (chain rule) to establish that $\frac{d}{dx}[e^{f(x)}]=f'(x)e^{e^{f(x)}}$
- use the composite function rule (chain rule) to establish that $\frac{d}{dx}[lnf(x)]=\frac{f'(x)}{f(x)}$
- use the logarithmic laws to simplify an expression before differentiating

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How to Differentiate Exponential Functions

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Building on our knowledge of differentiation from the year 11 course, we should know that there are three standard derivative formulas for exponential functions:

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

\frac{d}{dx}e^{ax+b}=ae^{ax+b}

\frac{d}{dx}e^u=e^u\frac{du}{dx}\quad or \quad \frac{d}{dx}e^{f(x)}=e^{f(x)}f'(x)

It should be noted that the base formula for deriving $e^x$ is also $e^x$ . However, when deriving the expression $e^{-x}$ , it is important to note that the negative sign is moved out the front, as follows: $-e^{-x}$ .

Practice Question 1

Derive the following function $f(x)=x^3e^{3x+1}$ using the product rule, and any related formulas provided above.

Solution

First we must determine the values of $u$ and $v$ .

u=x^3\qquad v=e^{3x+1}

Using differentiation formulas and techniques learnt so far, find $u'$ and $v'$ :

u'=3x^2\qquad v'=3e^{3x+1}

Finally, we can substitute these values into the product rule formula, to determine the derivative:

f'(x)=u'v+uv'\\=(3x^2\times e^{3x+1})+(x^3\times 3e^{3x+1})\\=3x^2e^{3x+1}+3x^3e^{3x+1}\\=3x^2e^{3x+1}(1+x)

How to Differentiate Logarithmic Functions

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To begin our understanding of differentiating logarithmic functions, it is important to start with the reciprocal function. This is a $log$ function of base $e$ denoted by the formula:

\frac{d}{dx}log_ex=\frac{1}{x}\qquad or\qquad \frac{d}{dx}lnx=\frac{1}{x}

Practice Question 2

Differentiate the function, $y=7lnx$

Solution

Applying the formula above we can simply differentiate:

\frac{d}{dx}7lnx=7\times\frac{1}{x}\\=\frac{7}{x}

Now that we have learnt the general formula for the reciprocal function, there are two other standard formulas, which are important for us to apply to various questions.

\frac{d}{dx}log_e(ax+b)=\frac{a}{ax+b}

\frac{d}{dx}log_eu=\frac{u'}{u}\qquad or\qquad \frac{d}{dx}log_ef(x)=\frac{f'(x)}{f(x)}

By manipulating any expression or equation to fit the form of the formulas above, $log$ functions with base $e$ can be easily solved.

Practice Question 3

Differentiate the function $f(x)=\frac{log_e(x)}{x}$ using both the quotient rule, and any related formulas provided above.

Solution

We must first determine terms $u$ and $v$ to use in the quotient rule:

u=log_e(x)\qquad v=x

Using differentiation formulas and techniques learnt so far, find $u'$ and $v'$ :

u'=\frac{1}{x}\qquad v'=1

Finally, we can substitute these values into the quotient rule formula, to determine the derivative:

f'(x)=\frac{u'v-uv'}{v^2}\\=\frac{\frac{1}{x}x-log_e(x)\times1}{x^2}\\=\frac{1-log_e(x)}{x^2}

Review of Differentiation and Log Functions

Integration of Exponential Functions

Return to Module 5: Exponential and Logarithmic Functions