Trigonometric Differentiation

Reference > Calculus: Differentiation

Description

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

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

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

$\frac{d}{dx} \csc{x}=-\csc{x}\cot{x}$

$\frac{d}{dx} \sec{x}=\sec{x}\tan{x}$

$\frac{d}{dx} \cot{x}=-\csc^{2}x$

Examples

Example 1

\frac{d}{dx} 3\sin{x}+7

Use Sum Rule:

\frac{d}{dx} f(x)+g(x)=(\frac{d}{dx} f(x))+(\frac{d}{dx} g(x))

(\frac{d}{dx} 3\sin{x})+(\frac{d}{dx} 7)

Use Constant Factor Rule:

\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x))

3(\frac{d}{dx} \sin{x})+(\frac{d}{dx} 7)

Use Trigonometric Differentiation: the derivative of

\sin{x}

\cos{x}

3\cos{x}+(\frac{d}{dx} 7)

Use this rule:

\frac{d}{dx} c=0

3\cos{x}

Done

Example 2

\frac{d}{dx} \sec^{2}x\cos{x}

Use Product Rule to find the derivative of

\sec^{2}x\cos{x}

. The product rule states that

(fg)'=f'g+fg'

(\frac{d}{dx} \sec^{2}x)\cos{x}+\sec^{2}x(\frac{d}{dx} \cos{x})

Use Chain Rule on

\frac{d}{dx} \sec^{2}x

. Let

u=\sec{x}

. Use Power Rule:

\frac{d}{du} {u}^{n}=n{u}^{n-1}

2\sec{x}(\frac{d}{dx} \sec{x})\cos{x}+\sec^{2}x(\frac{d}{dx} \cos{x})

Use Trigonometric Differentiation: the derivative of

\sec{x}

\sec{x}\tan{x}

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

Use Trigonometric Differentiation: the derivative of

\cos{x}

-\sin{x}

2\sec^{2}x\tan{x}\cos{x}-\sec^{2}x\sin{x}

Done

Example 3

\frac{d}{dx} \frac{\tan{x}}{4}+\sin{x}

Use Sum Rule:

\frac{d}{dx} f(x)+g(x)=(\frac{d}{dx} f(x))+(\frac{d}{dx} g(x))

(\frac{d}{dx} \frac{\tan{x}}{4})+(\frac{d}{dx} \sin{x})

Use Constant Factor Rule:

\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x))

\frac{1}{4}(\frac{d}{dx} \tan{x})+(\frac{d}{dx} \sin{x})

Use Trigonometric Differentiation: the derivative of

\tan{x}

\sec^{2}x

\frac{\sec^{2}x}{4}+(\frac{d}{dx} \sin{x})

Use Trigonometric Differentiation: the derivative of

\sin{x}

\cos{x}

\frac{\sec^{2}x}{4}+\cos{x}

Done