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Differentiating trig, exponentials and logs
The power rule ran Year 12; now the rest of the standard functions join the calculus. The derivative of sine is cosine, the derivative of ex is itself, and the derivative of ln x is 1/x. Each of these results depends on the angle being measured in radians.
Builds on Differentiating powers of x and Radians, arcs and small angles.
IN THIS TOPIC
- Differentiate sin kx, cos kx, tan kx, ekx and ln x fluently.
- Handle akx, whose derivative carries a factor of ln a.
- Differentiate sin x and cos x from first principles using the small-angle approximations.
- Find gradients and tangents on trig, exponential and log curves.
COMMON MISCONCEPTION
The derivative of sin x is cos x, whatever unit the angle is in.
The full derivative shelf
Four families join the power rule, all of them on the must-learn list.
Alongside them sits tan kx → k sec2 kx, which the formulae booklet supplies, and one entry the specification names in its own words. Differentiate akx and you get kakx ln a. The factor ln a appears for every base except e, where ln e = 1 and the factor disappears; omitting it is the most common lost mark in this topic.
WORKED EXAMPLE
A mixed bag, termwise
Differentiate y = 4 sin 3x − 2 cos 5x + e3x.
Termwise from the shelf: 4 sin 3x gives 12 cos 3x, and −2 cos 5x gives +10 sin 5x, the two minus signs cancelling.
The exponential gives 3e3x.
dy/dx = 12 cos 3x + 10 sin 5x + 3e3x.
Each k multiplies out front and stays put inside. Cosine's sign flip is the one detail separating five marks from three.
Why sine's derivative is cosine
The specification asks for sin x from first principles, and the proof is the radians lesson cashing its cheque. Expand the chord gradient with the compound angle formula, regroup, and the two small-angle facts finish it off.
WORKED EXAMPLE
sin x from first principles
Prove from first principles that the derivative of sin x is cos x, for x in radians.
The chord gradient is [sin (x + h) − sin x]/h = [sin x cos h + cos x sin h − sin x]/h.
Regroup: sin x (cos h − 1)/h + cos x (sin h/h).
As h → 0, the small-angle approximations give (cos h − 1)/h → 0 and sin h/h → 1, leaving cos x. ∎
Name every ingredient, the compound formula and then both limits. In degrees sin h/h approaches π/180 instead of 1, and the derivative picks up that factor.
Gradients on the new curves
Nothing about tangents changes. Read the gradient off the shelf, evaluate it, and hand the number to point-gradient form. What does change is the arithmetic, because exact values of sin and cos at multiples of π/6 now do most of the work, and a calculator that has quietly slipped into degrees will ruin an otherwise perfect solution.
GUIDED PRACTICE
A tangent on a trig curve
Find the equation of the tangent to y = sin 2x at the point where x = π/6, before opening the working.
Show the working
The point: y = sin (π/3) = √3/2.
The gradient: dy/dx = 2 cos 2x, which at x = π/6 is 2 cos (π/3) = 1.
Tangent: y − √3/2 = 1 × (x − π/6), that is y = x − π/6 + √3/2.
Almost everything here was exact values. The calculus amounted to one shelf lookup and one k multiplying out.
INDEPENDENT PRACTICE
A base that is not e
Find the gradient of y = 5 × 2x at x = 3.
Show the working
The shelf gives dy/dx = 5 × 2x ln 2.
At x = 3 the gradient is 5 × 8 × ln 2 = 40 ln 2 ≈ 27.7.
That ln 2 is what separates base 2 from base e. Only e's curve grows at exactly its own height, as the exponentials lesson promised, and every other base carries its logarithm as a correction.
ASSESSMENT FOCUS
- Radians throughout. The trig derivatives are false in degrees and the mark scheme knows it.
- The k multiplies out front and survives inside: sin 3x gives 3 cos 3x, both threes present.
- Cosine's derivative carries the minus sign. Write the shelf line down before substituting anything.
- First-principles proofs for sin x want the compound expansion, the regrouping, and both small-angle limits named.
- For akx, the derivative is kakx ln a. Forgetting the ln a is the standard lost mark.
CHECK YOURSELF
Differentiate y = 3 ln x − cos 4x, and find the gradient at x = π/4.
Show a hint
Termwise from the shelf; cos π has an exact value.
Show the answer
dy/dx = 3/x + 4 sin 4x.
At x = π/4: 3/(π/4) + 4 sin π = 12/π + 0 = 12/π ≈ 3.82.
The sine term vanished at a multiple of π. Exact-value fluency spots that before the calculator ever comes out.
Sine to cosine, cosine to minus sine, e to itself, ln to the reciprocal: the shelf, in radians.
Every k multiplies out front; every base other than e pays a factor of its log.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the differentiating trig, exponentials and logs questions page.
CHECK YOUR PROGRESS
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- Differentiate sin kx, cos kx, tan kx, ekx and ln x fluently.
- Handle akx, whose derivative carries a factor of ln a.
- Differentiate sin x and cos x from first principles using the small-angle approximations.
- Find gradients and tangents on trig, exponential and log curves.
Open the full revision checklist to see every objective in the course in one place.