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Differentiating trig, exponentials and logs

The remaining standard functions have their own derivatives: the derivative of sine is cosine, the derivative of ex is itself, and the derivative of ln x is 1/x. The trigonometric results take these forms only with the angle measured in radians, and the derivatives of sine and cosine can both be proved from first principles, using the compound angle formulae and the small-angle limits.

Year 12-13EDEXCEL 9MA0 7.1, 7.2

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 result holds with x in radians. The first-principles proof rests on sin h/h tending to 1 as h tends to zero, and with h in degrees that ratio tends to π/180 instead, so the derivative carries that factor. Work in radians throughout, and set the calculator to them before evaluating any gradient.

The standard derivatives

Four families of standard derivatives join the power rule, and all of them must be learnt. The trigonometric results hold with x in radians; the derivative of ln x is valid for x > 0, where ln x is defined.

sinkx→kcoskx,coskx→−ksinkx\sin kx → k\cos kx, \, \cos kx → −k\sin kxNOT IN THE BOOKLET: LEARN IT
ekx→kekx,ln x→1xe^{kx} → ke^{kx}, \, \text{ln } x → \frac{1}{x}NOT IN THE BOOKLET: LEARN IT

Alongside them, the formulae booklet gives tan kx → k sec2 kx, valid where cos kx is non-zero. One further result is named in the specification: akx differentiates to kakx ln a, for a > 0. For base e the factor is ln e = 1, so it disappears; for every other base, include it.

The sine wave with dashed verticals dropped from its peak and trough onto the cosine wave beneath, landing exactly on cosine's zeros: reading sine's slope point by point generates cosine, the fact the whole trig calculus rests on.
FIG. 1The derivative of sin x is cos x: the stationary points of sin x lie directly above the zeros of cos x. The result holds with x in radians.

WORKED EXAMPLE

Three standard derivatives, termwise

Differentiate y = 4 sin 3x − 2 cos 5x + e3x.

Termwise: 4 sin 3x gives 12 cos 3x, and −2 cos 5x gives +10 sin 5x, since the derivative of cos 5x is −5 sin 5x and the two minus signs cancel.

The exponential gives 3e3x.

dy/dx = 12 cos 3x + 10 sin 5x + 3e3x.

Each k multiplies the whole term and is unchanged inside the function. Keep the minus sign on the derivative of cosine.

The logarithm curve climbing ever more lazily, with amber tangent segments at 1, 2 and 4 whose computed slopes are 1, a half and a quarter: the derivative of ln x is 1 over x, falling as the curve flattens.
FIG. 2ln x with tangents at 1, 2 and 4: gradients 1, ½ and ¼. The derivative of ln x is 1/x for x > 0, so the gradient decreases as x increases.

Sine and cosine from first principles

Edexcel 9MA0 7.1 requires both sin x and cos x from first principles. Each proof expands the chord gradient with a compound angle formula, regroups the result into two parts, and uses the same pair of limits, sin h/h → 1 and (cos h − 1)/h → 0 as h → 0, with h in radians.

Both of those come from the booklet's small-angle approximations, with h in radians. Since sin h ≈ h, the first ratio is h/h, which is 1. Since cos h ≈ 1 − h2/2, the second is −h/2, and that shrinks away with h. The two proofs below need nothing beyond them.

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 each result used: the compound angle formula, then both limits. With h in degrees, sin h/h tends to π/180 rather than 1, and the derivative carries that factor.

WORKED EXAMPLE

cos x from first principles

Prove from first principles that the derivative of cos x is −sin x, for x in radians.

The chord gradient is [cos (x + h) − cos x]/h = [cos x cos h − sin x sin h − cos x]/h.

Regroup: cos x (cos h − 1)/h − sin x (sin h/h).

As h → 0 the same two limits give cos x × 0 − sin x × 1, that is −sin x. ∎

This is the sine proof with cos (A + B) supplying the expansion in place of sin (A + B). The minus sign in the derivative of cosine is the minus sign in that compound angle formula, carried through the regrouping.

Gradients on the new curves

The method for tangents is unchanged: take the standard derivative, evaluate it at the point, and use point-gradient form. The arithmetic uses exact values of sin and cos at multiples of π/6, and the calculator must be set to radians, since the derivatives above assume radian measure.

Three panels, one tangent each: the exponential curve touched at x equals 1 where the gradient is e, the logarithm touched at x equals 2 where the gradient is one half, and the sine curve touched at pi over 6 where the gradient is root 3 over 2, every slope checked against its shelf derivative.
FIG. 3One tangent on each of the three curves, with the gradient given by the standard derivative printed beneath it: the exponential touched at x = 1 with gradient e, ln x touched at x = 2 with gradient ½, and sin x touched at π/6 with gradient √3/2. Every slope is checked numerically against the drawn line.

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.

The calculus is one standard derivative with its factor of k; the rest is exact values.

INDEPENDENT PRACTICE

A base that is not e

Find the gradient of y = 5 × 2x at x = 3.

Show the working

The standard result gives dy/dx = 5 × 2x ln 2.

At x = 3 the gradient is 5 × 8 × ln 2 = 40 ln 2 ≈ 27.7.

The factor ln 2 distinguishes base 2 from base e. Only for base e is the derivative of the exponential equal to the function itself; every other base carries a factor of the natural logarithm of the base.

ASSESSMENT FOCUS

  • Work in radians throughout. The trigonometric derivatives above are not valid for x in degrees.
  • The k multiplies the term and remains inside the function: sin 3x gives 3 cos 3x.
  • The derivative of cos kx carries a minus sign. Write the standard derivative down before substituting.
  • A complete first-principles proof for sin x or cos x states the compound angle expansion, the regrouping, and both small-angle limits. The two proofs differ only in which compound angle formula opens them.
  • For akx with a > 0, the derivative is kakx ln a. Include the factor ln a for every base except e.

CHECK YOURSELF

Differentiate y = 3 ln x − cos 4x, and find the gradient at x = π/4.

Show a hint

Termwise from the standard derivatives; sin 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 is zero because sin π = 0, which the exact values give without a calculator.

With x in radians, sin x differentiates to cos x, cos x to −sin x, ex to itself, and ln x to 1/x for x > 0.

Every k multiplies the term; every base other than e carries a factor ln a.

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.

Question workbook

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Question workbook

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Answer book

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15 questions on this topicAnswer them one at a time and mark yourself against the worked answer.Practise this topic

Or read them with their worked answers on the differentiating trig, exponentials and logs questions page.

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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.