MCV4U Derivatives of Sine and Cosine: Notes and Practice

Tides, Ferris wheels, a swinging pendulum, the length of daylight over a year: all of these repeat, and you modelled them with sine and cosine functions. Calculus asks the next question. When is the tide rising fastest? How fast is a rider on a Ferris wheel climbing at a given moment? To answer, you need the derivative of $sin x$ and $cos x$ , and you can find both by looking carefully at a graph.

Tall reddish rock pillars topped with trees standing on an exposed muddy sea floor, with people walking among them and the water far off in the distance. — PhotoHopewell Rocks on the Bay of Fundy in New Brunswick at low tide, when people can walk on the sea floor. The tide here rises and falls by many metres twice a day, a repeating pattern that sine and cosine models describe, and derivatives tell you when it is rising fastest.Wladyslaw, CC BY-SA 3.0, via Wikimedia Commons

Reading slopes off the sine curve

Trace $y = sin x$ from $x = 0$ . At $0$ it climbs at its steepest, with tangent slope $1$ . At $\frac{π}{2}$ it reaches its peak, where the tangent is flat: slope $0$ . At $π$ it falls through the axis at its steepest: slope $- 1$ . At $\frac{3 π}{2}$ it bottoms out: slope $0$ . At $2 π$ it is climbing at slope $1$ again. The slopes $1, 0, - 1, 0, 1$ are exactly the values of $cos x$ at those points.

Fig 1Left column: y = sin x with tangent slopes 1, 0, −1, 0, 1 marked at 0, π/2, π, 3π/2 and 2π, and y = cos x passing through those values below it. Right column: y = cos x with tangent slopes 0, −1, 0, 1, 0 at the same points, and below it −sin x passing through those values.

\frac{d}{d x} (sin x) = cos x \frac{d}{d x} (cos x) = - sin x

Both rules require x to be measured in radians.

Loading interactive…

Choose f(x) = 2sin(x), pick a point, and shrink h. The secant slope closes in on the exact rate 2cos(x) at every point you try.

h	Secant slope of sin x from 0 to h, that is sin(h) divided by h
$0.1$	$0.99833$
$0.01$	$0.99998$
$0.001$	$1.00000$
Prediction from the rule	$cos 0 = 1$

Numerical check at x = 0 with h in radians, rounded to 5 decimal places.

Radians are not optional here

x

were measured in degrees, the sine curve would be stretched out over

360

units per cycle instead of

2 π

, and its slope at

0

would be only

\frac{π}{180} \approx 0.01745

, not

1

. The clean rules

\frac{d}{d x} sin x = cos x

and

\frac{d}{d x} cos x = - sin x

are true only in radians. Keep your calculator in RAD mode for every calculation in this lesson.

VideoSal Khan compares the slope of the sine curve at key points with the cosine curve, and does the same for cosine and negative sine, the graphical argument this section makes.Watch on YouTube

Combining with the other rules

Function	Derivative	Rule used
$sin (k x)$	$k cos (k x)$	Chain, inner $k x$
$cos (x^{2})$	$- 2 x sin (x^{2})$	Chain, inner $x^{2}$
$sin^{2} x = (sin x)^{2}$	$2 sin x cos x$	Chain, outer $u^{2}$
$x sin x$	$sin x + x cos x$	Product
$tan x = \frac{sin x}{cos x}$	$\frac{1}{cos ^{2} x}$	Quotient

The new rules plug into the old ones exactly like any other function.

The last row is worth doing once by hand. With the quotient rule, $\frac{d}{d x} (\frac{sin x}{cos x}) = \frac{cos x cos x - sin x ( - sin x )}{cos ^{2} x} = \frac{cos ^{2} x + sin ^{2} x}{cos ^{2} x} .$ The Pythagorean identity $sin^{2} x + cos^{2} x = 1$ turns the numerator into $1$ , so $\frac{d}{d x} tan x = \frac{1}{cos ^{2} x}$ , which is also written $sec^{2} x$ .

Differentiate $sin x$ repeatedly and the results cycle with period four: $sin x$ , then $cos x$ , then $- sin x$ , then $- cos x$ , and back to $sin x$ .
So the second derivative of $sin x$ is $- sin x$ : the second derivative is always the negative of the function, so the two have opposite signs wherever $sin x \neq = 0$ . This is why a mass bouncing on a spring (or a pendulum making small swings), whose acceleration always pulls back toward the centre, moves sinusoidally.
In a model $y = a sin (k (x - d)) + c$ , the vertical shift $c$ and the phase shift $d$ do not change the size of the slopes, but the amplitude $a$ and the factor $k$ both multiply them: the maximum rate of change is $∣ ak ∣$ .

Loading interactive…

Choose sin x or cos x, drag a point along the curve, and watch the derivative graph trace out cos x or negative sin x.

Sine squared is not sine of x squared

sin^{2} x

means

(sin x)^{2}

: the outer function is squaring, so its derivative is

2 sin x cos x

sin (x^{2})

means the sine of

x^{2}

: the outer function is sine, so its derivative is

cos (x^{2}) \cdot 2 x

. Before differentiating, decide which operation happens LAST; that is the outer function.

Derivatives of Sine and Cosine

Key things to remember

Reading slopes off the sine curve

Combining with the other rules

Worked examples

Where marks get lost

Flashcards 7

Practice