HSC Maths Advanced Syllabus

How to Differentiate Exponential Functions

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: $-e^{-x}$.

How to Differentiate Logarithmic Functions

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:

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

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.