Derivatives of sine/cosine/tangent | Year 12

01 / LEARN

Understand the idea.

Radians are part of the rule

For radian inputs, the derivatives of $sin x$ , $cos x$ and $tan x$ are $cos x$ , $- sin x$ and ${sec}^{2} x$ respectively, with $sec x = \frac{1}{cos x}$ . A numerical degree input would require an additional conversion factor.

A scaled angle needs the chain rule: $\frac{d}{d x} sin (k x) = k cos (k x)$ . Keep the same angle inside the trigonometric function; differentiating does not replace $k x$ by $x$ .

Structure and domain

Read the full structure before selecting a rule. A product needs the product rule, while ${sin}^{2} x = {(sin x)}^{2}$ needs the chain rule for an outer square and an inner sine.

Tangent and its derivative require a nonzero cosine. For $tan x$ , exclude $x = \frac{π}{2} + k π$ , $k \in ℤ$ ; for a composite angle, solve the corresponding cosine-zero condition for the actual input.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Scaled trigonometric angles

Differentiate $y = 3 cos (2 x) - sin x$ .

Worked solution

1
Differentiate the cosine term using its inner angle.
$\frac{d}{d x} (3 cos (2 x)) = 3 (- sin (2 x)) \cdot 2 = - 6 sin (2 x)$
Cosine contributes a negative sine and the angle contributes a factor 2.
2
Differentiate the remaining term.
$\frac{d}{d x} (- sin x) = - cos x$
The coefficient -1 remains attached to the sine derivative.
3
Combine the derivatives.
$y^{'} = - 6 sin (2 x) - cos x$
Differentiation acts term by term on a sum or difference.

Answer

$- 6 sin (2 x) - cos x$

EXAMPLE 02

A periodic height model

A height is $h (t) = 2 + sin (π t / 4)$ metres. Find its rate at $t = 0$ seconds.

Worked solution

1
Differentiate the height model.
$h^{'} (t) = \frac{π}{4} cos (\frac{π t}{4})$
The constant height offset has zero derivative; the angle rate is pi/4.
2
Substitute the stated time.
$h^{'} (0) = \frac{π}{4} cos 0 = \frac{π}{4}$
Cosine is 1 at zero radians.
3
Interpret with the correct units.
$h^{'} (0) = \frac{π}{4} m s^{- 1}$
The positive rate means height is increasing at the initial instant.

Answer

$\frac{π}{4}$ metres per second

03 / YOUR TURN

Try it yourself.

Check your understanding

Differentiate ${sin}^{2} x$ .

Show answer & reasoning

$2 sin x cos x$

Treat ${sin}^{2} x$ as an outer square of $sin x$ . The outer derivative gives $2 sin x$ and the inner derivative gives $cos x$ , so the result is $2 sin x cos x$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Differentiate $5 sin x + 2 cos x$ .
2 marks
Differentiate $tan (3 x)$ .
2 marks
Find the gradient of $y = cos (2 x)$ at $x = \frac{π}{4}$ .
2 marks
Differentiate $x sin x$ .
2 marks
Differentiate ${cos}^{3} x$ .
2 marks
For $h (t) = 5 + 2 cos (\frac{π t}{6})$ metres, find and interpret $h^{'} (3)$ in m/s.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Differentiate $5 sin x + 2 cos x$ .
2 marks
Apply the sine and cosine rules term by term.
$5 cos x - 2 sin x$
Differentiate $tan (3 x)$ .
2 marks
Use the tangent rule.
Multiply by 3 from the inner angle.
$3 {sec}^{2} (3 x)$
Find the gradient of $y = cos (2 x)$ at $x = \frac{π}{4}$ .
2 marks
The derivative is $- 2 sin (2 x)$ .
At $\frac{π}{4}$ , the sine is 1.
$- 2$
Differentiate $x sin x$ .
2 marks
Use the product rule.
Differentiate one factor at a time.
$sin x + x cos x$
Differentiate ${cos}^{3} x$ .
2 marks
Differentiate the outer cube.
Multiply by $- sin x$ .
$- 3 {cos}^{2} x sin x$
For $h (t) = 5 + 2 cos (\frac{π t}{6})$ metres, find and interpret $h^{'} (3)$ in m/s.
3 marks
Differentiate to get $- (\frac{π}{3}) sin (\frac{π t}{6})$ .
At $t = 3$ , the angle is $\frac{π}{2}$ .
$- \frac{π}{3}$ m/s: the height is decreasing.

Checking saved progress…

Curriculum reference & lesson scope

SACE Stage 2 Mathematical Methods

Further differentiation and applications from the current SACE Methods outline. Logarithmic derivatives have their own chapter; integration follows later. Trigonometric calculus uses radians.

Official curriculum source ↗

In this lesson

Understand the idea.

Radians are part of the rule

Structure and domain

See the method in action.

Scaled trigonometric angles

Differentiate the cosine term using its inner angle.

Differentiate the remaining term.

Combine the derivatives.

A periodic height model

Differentiate the height model.

Substitute the stated time.

Interpret with the correct units.

Try it yourself.

Check your understanding

Keep practising