Pure Mathematics 2

Mathematics9709

Pure Mathematics 2

Mathematics9709

Trigonometric Derivatives

These trigonometric derivative rules use radians. Radian measure makes the small-angle rate scale equal to 1; in degrees every formula would need an extra conversion factor.

The three derivatives

These three formulas assume that the input angle is in radians.

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

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

\frac{d}{dx}(\tan x)=\sec^2x

Secant is the reciprocal of cosine: $\sec x=1/\cos x$ , so $\sec^2x=1/\cos^2x$ . Cosine is the only one of the three rules above that begins with a minus sign.

Composite angles

Key idea

For an inside angle

u=u(x)

(\sin u)'=u'\cos u

(\cos u)'=-u'\sin u

and

(\tan u)'=u'\sec^2u

Worked example

Differentiate three composite angles

\frac{d}{dx}(2\sin3x)=6\cos3x

y=4\cos(2x-1)

\frac{dy}{dx}=-8\sin(2x-1)

\frac{d}{dx}(\tan5x)=5\sec^2 5x

Combine rules

Worked example

A product with cosine

y=x\cos x

y'=\cos x-x\sin x

Your turnIndependent check [5 marks]

Differentiate $y=(3-2\cos5x)^4$ and simplify.

Show worked answer

(3-2\cos5x)'=10\sin5x

Let $u=3-2\cos5x$ so the two factors stay readable on a narrow screen.

Answer

y'=40\sin5x\,u^3

u=3-2\cos5x

PreviousComposite Logarithmic Functions NextMixed Rules and Stationary Points

Trigonometric Derivatives

These trigonometric derivative rules use radians. Radian measure makes the small-angle rate scale equal to 1; in degrees every formula would need an extra conversion factor.

The three derivatives

These three formulas assume that the input angle is in radians.

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

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

\frac{d}{dx}(\tan x)=\sec^2x

Secant is the reciprocal of cosine: $\sec x=1/\cos x$ , so $\sec^2x=1/\cos^2x$ . Cosine is the only one of the three rules above that begins with a minus sign.

Composite angles

Key idea

For an inside angle

u=u(x)

(\sin u)'=u'\cos u

(\cos u)'=-u'\sin u

and

(\tan u)'=u'\sec^2u

Worked example

Differentiate three composite angles

\frac{d}{dx}(2\sin3x)=6\cos3x

y=4\cos(2x-1)

\frac{dy}{dx}=-8\sin(2x-1)

\frac{d}{dx}(\tan5x)=5\sec^2 5x

Combine rules

Worked example

A product with cosine

y=x\cos x

y'=\cos x-x\sin x

Your turnIndependent check [5 marks]

Differentiate $y=(3-2\cos5x)^4$ and simplify.

Show worked answer

(3-2\cos5x)'=10\sin5x

Let $u=3-2\cos5x$ so the two factors stay readable on a narrow screen.

Answer

y'=40\sin5x\,u^3

u=3-2\cos5x

PreviousComposite Logarithmic Functions NextMixed Rules and Stationary Points