Differentiate $y = \sin^2(3x)$ and simplify.

$y' = 6\sin(3x)\cos(3x) = 3\sin(6x)$

Find the absolute maximum and minimum of $f(x) = x + 2\cos x$ on $[0, \pi]$.

Absolute max $\frac{\pi}{6} + \sqrt{3} \approx 2.26$ at $x = \frac{\pi}{6}$; absolute min $\frac{5\pi}{6} - \sqrt{3} \approx 0.89$ at $x = \frac{5\pi}{6}$.

Grade 12 · MCV4U · Calculus & Vectors

MCV4U: Derivatives of Sin, Cos and Tan

The derivative of $\sin x$ is $\cos x$ , and the derivative of $\cos x$ is $-\sin x$ . Every other trig derivative in MCV4U comes from those two and the rules you already know.

How It Works

The three rules.

$\frac{d}{dx}\sin x = \cos x, \qquad \frac{d}{dx}\cos x = -\sin x, \qquad \frac{d}{dx}\tan x = \sec^2 x$

You can see the first one on the graphs. Where $y = \sin x$ peaks, its slope is $0$ , and that is where $\cos x = 0$ . Where $\sin x$ crosses the axis going up, its slope is steepest, $1$ , and $\cos x = 1$ there.

The tangent rule comes from the quotient rule: $\tan x = \frac{\sin x}{\cos x}$ , so its derivative is $\frac{\cos x \cos x - \sin x(-\sin x)}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$ , using $\sin^2 x + \cos^2 x = 1$ .

Radians only. These rules hold when $x$ is in radians. In degrees every derivative would carry an extra factor of $\frac{\pi}{180}$ . Set your calculator to radians before evaluating.

With the chain rule. Multiply by the derivative of the angle: $\frac{d}{dx}\sin(kx) = k\cos(kx)$ . For a power of a trig function, $\sin^2 x$ means $(\sin x)^2$ , so the outside function is the square: $\frac{d}{dx}\sin^2 x = 2\sin x\cos x$ .

Sign check. The minus sign belongs to the derivative of cosine. Mixing that up is the most common error, so say the cycle to yourself: sine to cosine, cosine to negative sine.

Extreme values. Setting a trig derivative to zero gives a trig equation. Solve it on the interval the question gives, then compare the function at each solution and at the endpoints to find the absolute maximum and minimum.

Leave answers exact, with $\pi$ and roots, unless a decimal is asked for.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Differentiate $y = 3\sin x - 2\cos x$ .
Show solution
1. $\frac{d}{dx}3\sin x = 3\cos x$ and $\frac{d}{dx}(-2\cos x) = 2\sin x$ .
Answer: $y' = 3\cos x + 2\sin x$
Easy
Differentiate $y = \sin(4x) - \cos(2x)$ .
Show solution
1. Chain rule on each: $\frac{d}{dx}\sin(4x) = 4\cos(4x)$ .
2. $\frac{d}{dx}(-\cos(2x)) = 2\sin(2x)$ .
Answer: $y' = 4\cos(4x) + 2\sin(2x)$
Medium
Differentiate $y = \sin^2(3x)$ and simplify.
Show solution
1. Rewrite as $y = (\sin(3x))^2$ . Outside: the square. Inside: $\sin(3x)$ , with derivative $3\cos(3x)$ .
2. $y' = 2\sin(3x) \cdot 3\cos(3x) = 6\sin(3x)\cos(3x)$ .
3. Double angle formula: $2\sin\theta\cos\theta = \sin(2\theta)$ , so $y' = 3\sin(6x)$ .
Answer: $y' = 6\sin(3x)\cos(3x) = 3\sin(6x)$
Medium
Find the equation of the tangent line to $y = \tan x$ at $x = \frac{\pi}{4}$ .
Show solution
1. Point: $\tan\frac{\pi}{4} = 1$ .
2. Slope: $\sec^2\frac{\pi}{4} = \frac{1}{\cos^2\frac{\pi}{4}} = \frac{1}{\frac{1}{2}} = 2$ .
3. $y - 1 = 2\left(x - \frac{\pi}{4}\right)$ , so $y = 2x + 1 - \frac{\pi}{2}$ .
Answer: $y = 2x + 1 - \frac{\pi}{2}$
Test level
Find the absolute maximum and minimum of $f(x) = x + 2\cos x$ on $[0, \pi]$ .
Show solution
1. $f'(x) = 1 - 2\sin x = 0$ gives $\sin x = \frac{1}{2}$ , so $x = \frac{\pi}{6}$ or $x = \frac{5\pi}{6}$ in the interval.
2. Endpoints: $f(0) = 2$ and $f(\pi) = \pi - 2 \approx 1.14$ .
3. Critical numbers: $f\left(\frac{\pi}{6}\right) = \frac{\pi}{6} + \sqrt{3} \approx 2.26$ and $f\left(\frac{5\pi}{6}\right) = \frac{5\pi}{6} - \sqrt{3} \approx 0.89$ .
4. Compare all four values.
Answer: Absolute max $\frac{\pi}{6} + \sqrt{3} \approx 2.26$ at $x = \frac{\pi}{6}$ ; absolute min $\frac{5\pi}{6} - \sqrt{3} \approx 0.89$ at $x = \frac{5\pi}{6}$ .

Learn It on Martian Lab

Martian Lab teaches this in the Derivatives of Sinusoidal Functions lesson of its MCV4U Derivatives of Trig, Exp and Log unit. Watch it, then take the Derivatives of Trig, Exp and Log practice tests to see if it stuck.

Watch the Derivatives of Sinusoidal Functions lesson free

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Free Derivatives of Trig, Exp and Log practice test, with solutions

More Derivatives of Trig, Exp and Log topics: Derivatives of e^x and a^x

More MCV4U topics: Evaluating Limits · Chain Rule · Optimization Problems · Related Rates · Curve Sketching Steps · Dot Product and Projections · Cross Product, Area and Torque · Equations of Planes · Intersection of Three Planes · Continuity · Derivative from First Principles · Power, Product and Quotient Rules · Velocity and Acceleration · Increasing, Decreasing and Critical Points · Concavity and the Second Derivative Test · Adding and Subtracting Vectors · Cartesian Vectors in 2D and 3D · Force and Velocity Vector Problems · Equations of Lines in 3D · Intersection of Lines and Planes · Distance from a Point to a Line or Plane

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