QCE Maths Methods: calculus of exponential and logarithmic functions

The exponential function e^x is its own derivative, and ln(x) is its inverse. Two rules carry the topic: the derivative of e^f(x) is f'(x) e^f(x), and the derivative of ln(f(x)) is f'(x)/f(x).

External assessment	One examination worth 50%, set and marked by the QCAA
Paper 1	Technology-free, 25%: short response and multiple choice
Paper 2	Technology-active, 25%: short response and multiple choice
Time	5 minutes perusal and 90 minutes working time for each paper; the formula book is provided
Internal assessment	Set by your school: a modelling task 20%, two short response examinations 15% each

The ideas that carry the topic

The graph of y = e^x passes through (0, 1), is always increasing and has the x-axis as an asymptote. The graph of y = ln(x) is its reflection in y = x: it passes through (1, 0) and has the y-axis as an asymptote.
Because e^x and ln(x) are inverses, e^ln(x) = x for x > 0 and ln(e^x) = x for every x. So e^(2x) = 5 gives 2x = ln(5) and x = ln(5)/2.
With the chain rule, the derivative of e^(3x) is 3e^(3x), and the derivative of ln(x^2 + 1) is 2x/(x^2 + 1).
The derivative of ln(kx) is 1/x for any positive constant k, because ln(kx) = ln(k) + ln(x).

Where it is tested

Both papers can test any part of Units 3 and 4. Without a calculator, leave answers such as ln(5)/2 exact; with one, give the accuracy the question asks for.

Try 3 questions

Paper 1 · Multiple choice · 1 mark · Calculus of exponential and logarithmic functions
If $y = e^{x^{2} - 3 x}$ , then $\dfrac{dy}{dx}$ is
Answer it, every step marked
Paper 1 · Short response · 4 marks · Calculus of exponential and logarithmic functions
Answer each part without technology.
Answer it, every step marked
Paper 2 · Multiple choice · 1 mark · Calculus of exponential and logarithmic functions
The table shows values of $\dfrac{a^{h} - 1}{h}$ for $h = 0.0001$ , correct to four decimal places. $\begin{array}{c|cccc} a & 2.5 & 2.7 & 2.8 & 3.0 \\ \hline \dfrac{a^{h} - 1}{h} & 0.9163 & 0.9933 & 1.0297 & 1.0987 \end{array}$ Based on the table, which interval contains the value of $a$ for which $\displaystyle\lim_{h \to 0} \dfrac{a^{h} - 1}{h} = 1$ ?
Answer it, every step marked

Next step

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Questions people ask

Is ln(x) the same as log_e(x)?

Yes. ln(x) is the natural logarithm, the logarithm to base e. In our practice, type ln(x) or log_e(x); a bare log is read as base 10, as on a calculator.

Which unit is this topic in?

Unit 3, Topic 1: Differentiation of exponential and logarithmic functions.

Can this topic appear in either paper?

Yes. Both papers can test any subject matter from Units 3 and 4.