The derivatives of the trigonometric functions

Calculus, grade 11 (5 units) Lesson 22 of 26Learning

The derivatives of the trigonometric functions

The limit of

\displaystyle \frac{\sin x}{x}

When $\displaystyle x$ is in radians and approaches 0, the quotient $\displaystyle \frac{\sin x}{x}$ approaches 1: for a small angle, its sine is almost equal to the angle itself.

$\displaystyle x$	$\displaystyle 0.5$	$\displaystyle 0.1$	$\displaystyle 0.01$
$\displaystyle \frac{\sin x}{x}$	$\displaystyle 0.9589$	$\displaystyle 0.99833$	$\displaystyle 0.99998$

The reason, on the unit circle, when $\displaystyle 0<x<\frac{\pi}{2}$ :

The arc opposite the angle $\displaystyle x$ is $\displaystyle x$ long, and the perpendicular from the end of the arc to the $\displaystyle x$ -axis is $\displaystyle \sin x$ long – shorter than the arc: $\displaystyle \sin x<x$ .
The tangent to the circle at the point $\displaystyle (1,0)$ , up to the ray of the angle, is $\displaystyle \tan x$ long, and the triangle it closes has a greater area than the sector: $\displaystyle \frac{\tan x}{2}>\frac{x}{2}$ , that is $\displaystyle x<\tan x$ .
From $\displaystyle \sin x<x<\frac{\sin x}{\cos x}$ comes $\displaystyle \cos x<\frac{\sin x}{x}<1$ . As $\displaystyle x\to0$ , $\displaystyle \cos x\to1$ , so the quotient in the middle approaches 1 too.
$\displaystyle \frac{\sin x}{x}$ is even, so the same holds as $\displaystyle x$ approaches 0 from the negative side.

\displaystyle \lim_{x\to0}\frac{\sin x}{x}

\displaystyle =

\displaystyle 1

The derivative of

\displaystyle \sin x

The proof below works out the derivative of $\displaystyle \sin x$ by the definition, with an identity from the formula sheet:

\displaystyle \sin\alpha-\sin\beta

\displaystyle =

\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}

With $\displaystyle {\alpha}={x+h}$ and $\displaystyle {\beta}={x}$ : $\displaystyle {\sin(x+h)-\sin x}={2\cos\left(x+\frac{h}{2}\right)\sin\frac{h}{2}}$ .
Dividing by $\displaystyle h$ : $\displaystyle {\frac{\sin(x+h)-\sin x}{h}}={\cos\left(x+\frac{h}{2}\right)\cdot\frac{\sin\frac{h}{2}}{\frac{h}{2}}}$ .
As $\displaystyle h\to0$ : $\displaystyle \cos\left(x+\frac{h}{2}\right)\to\cos x$ , and $\displaystyle \frac{\sin\frac{h}{2}}{\frac{h}{2}}\to1$ by the limit above.

\displaystyle (\sin x)'

\displaystyle =

\displaystyle \cos x

The formula fits the graphs: where $\displaystyle \sin x$ rises fastest (at $\displaystyle {x}={0}$ ) $\displaystyle \cos x$ is greatest, and at the extremum points of $\displaystyle \sin x$ ( $\displaystyle {x}={\frac{\pi}{2}+\pi k}$ ) $\displaystyle {\cos x}={0}$ .

Why radians

The limit $\displaystyle \frac{\sin x}{x}\to1$ holds only when the angle is in radians: it comes from the length of an arc of the unit circle being equal to the angle.

The sine of $\displaystyle x$ degrees is $\displaystyle \sin\frac{\pi x}{180}$ . By the chain rule its derivative is $\displaystyle \frac{\pi}{180}\cos\frac{\pi x}{180}$ – with an extra factor $\displaystyle \frac{\pi}{180}$ .
So in calculus $\displaystyle x$ is always in radians, and $\displaystyle {(\sin x)'}={\cos x}$ with no coefficient.

The derivatives of

\displaystyle \cos x

and of

\displaystyle \tan x

The other two derivatives follow from the derivative of $\displaystyle \sin x$ :

$\displaystyle {\cos x}={\sin\left(\frac{\pi}{2}-x\right)}$ . By the chain rule: $\displaystyle {(\cos x)'}={\cos\left(\frac{\pi}{2}-x\right)\cdot(-1)}={-\sin x}$ .
$\displaystyle {\tan x}={\frac{\sin x}{\cos x}}$ . By the quotient rule: $\displaystyle {(\tan x)'}={\frac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^2x}}={\frac{\cos^2x+\sin^2x}{\cos^2x}}={\frac{1}{\cos^2x}}$ .

\displaystyle (\cos x)'

\displaystyle =

\displaystyle -\sin x

\displaystyle (\tan x)'

\displaystyle =

\displaystyle \frac{1}{\cos^2x}

$\displaystyle (\tan x)'$ is positive throughout the domain: $\displaystyle \tan x$ rises on every interval between two asymptotes.

The chain rule

With the chain rule you differentiate a trigonometric function of an expression, and an expression of a trigonometric function:

$\displaystyle {(\sin3x)'}={\cos3x\cdot3}={3\cos3x}$ .
$\displaystyle {(\cos^2x)'}={2\cos x\cdot(-\sin x)}={-2\sin x\cos x}={-\sin2x}$ .
$\displaystyle {\left(\sqrt{\sin x}\right)'}={\frac{\cos x}{2\sqrt{\sin x}}}$ , where $\displaystyle \sin x>0$ .
$\displaystyle {\left(\sin(x^2)\right)'}={\cos(x^2)\cdot2x}$ – careful: $\displaystyle \sin(x^2)$ is not $\displaystyle {\sin^2x}={(\sin x)^2}$ .

Example: tangents to the graph of

\displaystyle \sin x

The example below finds the tangents to the graph of $\displaystyle {f(x)}={\sin x}$ at the points where $\displaystyle {x}={0}$ and $\displaystyle {x}={\pi}$ :

At $\displaystyle {x}={0}$ : $\displaystyle {f(0)}={0}$ and $\displaystyle {f'(0)}={\cos0}={1}$ , and the tangent is $\displaystyle {y}={x}$ .
At $\displaystyle {x}={\pi}$ : $\displaystyle {f(\pi)}={0}$ and $\displaystyle {f'(\pi)}={\cos\pi}={-1}$ , and the tangent is $\displaystyle {y-0}={-1(x-\pi)}$ , that is $\displaystyle {y}={\pi-x}$ .

\displaystyle f(x)=\sin x

\displaystyle y=x

\displaystyle y=\pi-x

Near $\displaystyle {x}={0}$ the graph of $\displaystyle \sin x$ almost coincides with the tangent $\displaystyle {y}={x}$ – the limit $\displaystyle \frac{\sin x}{x}\to1$ again.

Exercise 1

Differentiate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The function is $\displaystyle {f(x)}={\sin x+\cos x}$ .

Work out $\displaystyle f'\left(\frac{\pi}{4}\right)$ .
What can you conclude about the tangent to the graph at this point?

$\displaystyle f(x)=\sin x+\cos x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write on the board the equation of the tangent to the graph of $\displaystyle f$ at the point where $\displaystyle {x}={0}$ .

\displaystyle f(x)=\sin x+\cos x

Exercise 3

The function is $\displaystyle {f(x)}={\tan x}$ , with the point $\displaystyle \left(\frac{\pi}{4},1\right)$ on its graph.

Work out the slope of the tangent at the point.

$\displaystyle f(x)=\tan x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write the equation of the tangent on the board.

\displaystyle f(x)=\tan x

\displaystyle P=(\frac{\pi}{4},1)

Exercise 4

The function $\displaystyle {f(x)}={x+2\cos x}$ is defined on the interval $\displaystyle 0\leq x\leq2\pi$ .
You are looking for the points where the tangent to the graph is parallel to the $\displaystyle x$ -axis.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x+2\cos x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ in the interval.

Mark the points on the graph.

\displaystyle f(x)=x+2\cos x

Exercise 5

The function is $\displaystyle {f(x)}={\sin2x}$ .

Find the second derivative $\displaystyle f''(x)$ .

$\displaystyle f(x)=\sin2x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Show that $\displaystyle {f''(x)}={-4f(x)}$ for every $\displaystyle x$ .
What can you conclude about the concavity of $\displaystyle f$ where it is positive?

Formulas: calculus, grade 11 (5 units)

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Rules of differentiation and the tangent line

The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a product
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The tangent line at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle y-y_1=f'(x_1)(x-x_1)$

The second derivative

The second derivative – the derivative of the derivative
$\displaystyle f''(x)$ $\displaystyle =$ $\displaystyle \left(f'(x)\right)'$
The function is concave up ( $\displaystyle \cup$ )
$\displaystyle f''(x)>0$
The function is concave down ( $\displaystyle \cap$ )
$\displaystyle f''(x)<0$
A possible inflection point – check that the sign of $\displaystyle f''$ changes there
$\displaystyle f''(x_0)=0$
A minimum point
$\displaystyle f'(x_0)=0,\ f''(x_0)>0$
A maximum point
$\displaystyle f'(x_0)=0,\ f''(x_0)<0$
Acceleration – the derivative of the speed
$\displaystyle a(t)=v'(t)=s''(t)$

Asymptotes

A vertical asymptote – $\displaystyle f(x)$ grows or falls without end as $\displaystyle x$ approaches $\displaystyle a$
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle f(x)$ approaches $\displaystyle b$ as $\displaystyle x\to\infty$ or as $\displaystyle x\to-\infty$
$\displaystyle y=b$

Roots

The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Radians and periods

Half a turn in radians
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians in a circle of radius $\displaystyle R$
$\displaystyle l$ $\displaystyle =$ $\displaystyle \alpha R$
The period of $\displaystyle \sin x$ and of $\displaystyle \cos x$ is $\displaystyle 2\pi$
$\displaystyle \sin(x+2\pi)$ $\displaystyle =$ $\displaystyle \sin x$
The period of $\displaystyle \tan x$ is $\displaystyle \pi$
$\displaystyle \tan(x+\pi)$ $\displaystyle =$ $\displaystyle \tan x$
The limit behind the derivative of $\displaystyle \sin x$ ( $\displaystyle x$ in radians)
$\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1$

Trigonometric identities

The Pythagorean identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$
Sine – an odd function
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
Cosine – an even function
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
The sine of the supplement to $\displaystyle \pi$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The cosine of the complement to $\displaystyle \frac{\pi}{2}$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The sine of a double angle
$\displaystyle \sin2\alpha$ $\displaystyle =$ $\displaystyle 2\sin\alpha\cdot\cos\alpha$
The cosine of a double angle
$\displaystyle \cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$
The difference of two sines
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =$ $\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}$
The sine of a sum and of a difference
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta$

Derivatives of the trigonometric functions

The derivative of sine ( $\displaystyle x$ in radians)
$\displaystyle (\sin x)'$ $\displaystyle =$ $\displaystyle \cos x$
The derivative of cosine
$\displaystyle (\cos x)'$ $\displaystyle =$ $\displaystyle -\sin x$
The derivative of tangent
$\displaystyle (\tan x)'$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2x}$