IB Maths AA SL
Topic 5 β Calculus
Paper 1 & 2
~10 min read
Differentiating Special Functions
So far we’ve only differentiated powers of x. But IB exams will also throw trig functions, exponentials, and logarithms at you. The good news: each of these has its own little rule, and they’re all in the formula booklet. Memorise four standard derivatives and the patterns that follow, and you can handle anything they ask.
π What you need to know
- The four standard results (memorise these):
y = sin x β yβ² = cos x Β·
y = cos x β yβ² = βsin x Β·
y = ex β yβ² = ex Β·
y = ln x β yβ² = 1x.
- For “ax + b” inside any of these, multiply the derivative by a: e.g. d/dx(sin(3x + 2)) = 3 cos(3x + 2).
- For ln(ax), the answer is just 1x β the a‘s cancel!
- Always work in radians when differentiating trig functions. Switch your GDC to radians mode straight away.
- For more complex inside functions f(x), the chain rule gives you the general result (next note).
- All four standard derivatives are in the formula booklet β no need to memorise the formulas if you know where to look.
The four standard derivatives
Here’s the foundation. Each of these stands on its own β you’ll use them all the time.
Sine
y = sin x
β differentiate
yβ² = cos x
Cosine
y = cos x
β differentiate
yβ² = βsin x
Exponential
y = ex
β differentiate
yβ² = ex
Natural log
y = ln x
β differentiate
yβ² = 1/x
π€ Why is ex its own derivative?
ex is the only function (apart from the trivial f = 0) that’s its own derivative. That’s exactly why e is the special constant it is β it’s defined to make this work. The number e β 2.718 is whatever value gives this perfect property.
As soon as you see a question with sin or cos, switch your GDC to radians mode. The trig derivatives only work in radians β using degrees here is the most common error in this whole topic.
Linear “inside” functions: ax + b
Real exam questions rarely give you a clean sin x or ex. They’ll usually have something like sin(3x + 2) or eβ2x instead. The pattern: the derivative gets multiplied by the coefficient of x. Same shape, just with an extra a tagged on the front.
| Function | Derivative | Notice |
|---|
| y = sin(ax + b) | yβ² = a cos(ax + b) | “a” jumps out front |
| y = cos(ax + b) | yβ² = βa sin(ax + b) | “a” jumps out, sign flips |
| y = eax + b | yβ² = a eax + b | “a” jumps out, e stays |
| y = ln(ax + b) | yβ² = a / (ax + b) | “a” goes on top |
π§ “The coefficient pops out front”
Whatever number is multiplying the x inside the function, that number jumps out and multiplies the whole derivative. The “+ b” sits along for the ride β it doesn’t affect the derivative because constants vanish when you differentiate.
The ln(ax) trick β the “a”s cancel!
This is a classic exam trap. When you see something like ln(5x) or ln(7x), it looks like the answer should involve a 5 or a 7 β but it doesn’t.
From the rule above with b = 0:
d/dx[ln(ax)] = a / (ax) = 1/x
The a‘s cancel! So d/dx(ln 5x) = 1/x, not 5/x. And d/dx(ln 100x) is also 1/x.
β Wrong
d/dx(ln 5x) = 5/x
treating the 5 like it has to appear somewhere
β Right
d/dx(ln 5x) = 1/x
a’s cancel: a/(ax) = 1/x
π€ Why do they cancel?
Log laws: ln(5x) = ln 5 + ln x. The first term is a constant β it differentiates to 0. The second term differentiates to 1/x. So d/dx[ln 5x] = 0 + 1/x = 1/x. The 5 was always going to vanish.
Two patterns to watch out for
Beyond the ln trap, there’s another classic error to avoid β applying the wrong rule to ekx.
β Wrong
d/dx(ekx) = kxΒ·ekx β 1
applying the power rule (this isn’t a power!)
β Right
d/dx(ekx) = kΒ·ekx
e to a function β coefficient pops out
πDon’t confuse ekx with xk
xk is “x raised to a constant” β use the power rule. ekx is “a constant raised to x” β use the exponential rule. They look similar but follow completely different rules.
What about more complex inside functions?
If the inside function isn’t just ax + b but something more complicated like x2 + 3 or 2x3, the same idea works β just multiply by the derivative of that inside function. This is the chain rule, and it’s the topic of the next note. For reference, here are the five general results:
| If y = | Then yβ² = |
|---|
| (f(x))n | n fβ²(x) (f(x))nβ1 |
| ef(x) | fβ²(x) ef(x) |
| ln(f(x)) | fβ²(x) / f(x) |
| sin(f(x)) | fβ²(x) cos(f(x)) |
| cos(f(x)) | βfβ²(x) sin(f(x)) |
Notice the pattern: the derivative of the outside function stays the same β just multiply it by fβ²(x) (the derivative of whatever’s inside). For now, stick to the linear-only versions; the next note covers the chain rule properly.
Worked examples
WE 1Differentiate trig functions
Find fβ²(x) for each:
(i) f(x) = sin x Β· (ii) f(x) = cos 2x Β· (iii) f(x) = 3 sin 4x β cos(2x β 3)
Apply the standard results. For each linear “inside”, the coefficient pops out front.part (i)
fβ²(x) = cos x
cos xpart (ii)
a = 2, so: fβ²(x) = β2 sin 2x
β2 sin 2xpart (iii)
Differentiate term by term:
3 sin 4x β 3 Γ 4 cos 4x = 12 cos 4x
βcos(2x β 3) β β(β2 sin(2x β 3)) = 2 sin(2x β 3)
fβ²(x) = 12 cos 4x + 2 sin(2x β 3)
double negative on (iii) β be very careful with the signs!
WE 2Find a gradient β exact value with trig
Find the gradient of the tangent to the curve y = sin(2x + Ο/6) at the point where x = Ο/8. Give your answer as an exact value.
Differentiate, then substitute. Keep angles in radians throughout.step 1 β differentiate
a = 2, so: dy/dx = 2 cos(2x + Ο/6)step 2 β substitute x = Ο/8
dy/dx = 2 cos(2(Ο/8) + Ο/6)
= 2 cos(Ο/4 + Ο/6) = 2 cos(5Ο/12)step 3 β exact value
cos(5Ο/12) = cos(Ο/4 + Ο/6) = cos(Ο/4)cos(Ο/6) β sin(Ο/4)sin(Ο/6)
= (β2/2)(β3/2) β (β2/2)(1/2) = (β6 β β2)/4
β 2 Γ (β6 β β2)/4 = (β6 β β2)/2
gradient = Β½(β6 β β2)
remember: GDC must be in RADIANS for trig calculus!
WE 3Differentiate exponentials and logs
Find dy/dx for: (i) y = e4x Β· (ii) y = ln(7x) Β· (iii) y = ln(2x β 5)
Apply the standard linear-inside rules.part (i)
a = 4: dy/dx = 4 e4x
4 e4xpart (ii) β the trick!
a = 7, b = 0 β a’s cancel:
dy/dx = 7/(7x) = 1/x
1/xpart (iii)
a = 2, b = β5: dy/dx = 2/(2x β 5)
2/(2x β 5)
part (ii) is the classic ln(ax) trap β don’t write 7/x!
WE 4Combined ex and ln β gradient at a point
A curve has equation y = eβ3x + 1 + 2 ln 5x. Find the gradient of the curve at the point where x = 2, giving your answer in the form a + bec, where a, b, c are integers.
Differentiate term by term. Watch the ln(5x) trick on the second term!step 1 β differentiate
eβ3x+1 β β3 eβ3x+1
2 ln 5x β 2 Γ (1/x) = 2/x (a’s cancel!)
dy/dx = β3 eβ3x+1 + 2/xstep 2 β substitute x = 2
dy/dx = β3 eβ3(2)+1 + 2/2
= β3 eβ5 + 1 = 1 β 3 eβ5
gradient = 1 β 3 eβ5, a = 1, b = β3, c = β5
if you wrote 10/x for the second term, you fell for the ln(5x) trap!
WE 5Mixed: trig, exp, and log together
Find fβ²(x) for f(x) = 4 cos(3x) β 2 eβx + ln(2x).
Differentiate term by term. Apply the right rule to each.term 1: 4 cos(3x)
a = 3: β 4 Γ (β3 sin(3x)) = β12 sin(3x)term 2: β2 eβx
a = β1: β β2 Γ (β1) eβx = 2 eβxterm 3: ln(2x)
a’s cancel: β 1/xCombine:
fβ²(x) = β12 sin(3x) + 2 eβx + 1/x
three different rules, three different terms β they don’t interact!
π‘ Top tips
- Switch your GDC to radians mode the moment you see sin or cos. Don’t wait until the end.
- Memorise the four standard results by heart β sin β cos, cos β βsin, ex β ex, ln x β 1/x. They’re in the formula booklet, but instant recall saves time.
- “a” pops out front” for any linear inside (ax + b). It applies to all four functions.
- For ln(ax), the answer is just 1/x β the a’s cancel. Always.
- Differentiate term by term for sums and differences. Each term follows its own rule independently.
- Watch the signs. cos differentiates to negative sin. Two negatives in (ii) of WE 1 became a positive β easy to slip on.
- Don’t confuse ekx with xk. The first uses the exponential rule, the second the power rule.
- For exact values with trig, recall that cos(Ο/4) = β2/2, sin(Ο/6) = 1/2, etc. These come up a lot.
β Common mistakes
- Working in degrees instead of radians. The trig derivatives are wrong by a factor of Ο/180 if you’re in degrees mode. Always switch first.
- The ln(kx) trap. Writing d/dx(ln 7x) = 7/x. Correct answer is 1/x β the a’s cancel.
- The ekx trap. Writing d/dx(e3x) = 3x Β· e3x β 1 by treating it like a power. Correct is 3 e3x.
- Sign error on cos. d/dx(cos x) = βsin x, NOT sin x. The negative is essential.
- Forgetting the “a” for linear-inside functions. d/dx(sin 5x) = 5 cos 5x, not just cos 5x.
- Adding instead of multiplying. The “a” multiplies the whole derivative; it doesn’t add to it.
- Confusing d/dx(ln x) with d/dx(log x). The natural log (ln) gives 1/x. The base-10 log gives 1/(x ln 10). At AA SL it’s almost always ln.
- Trying to apply chain rule too early. For linear-inside functions, the “a-pops-out” rule is enough β you don’t need full chain rule until the next note.
You’ve got the four standard derivatives plus the linear-inside extension. The next note brings in the chain rule properly β the master tool for differentiating any “function of a function”, not just linear insides. Once chain rule is in your toolkit, every general result on this page becomes a single, clean technique.
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