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 [Top]

\[\frac{d}{dx} 3\sin{x}+7\]
1
Use Sum Rule: \(\frac{d}{dx} f(x)+g(x)=(\frac{d}{dx} f(x))+(\frac{d}{dx} g(x))\).
\[(\frac{d}{dx} 3\sin{x})+(\frac{d}{dx} 7)\]

2
Use Constant Factor Rule: \(\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x))\).
\[3(\frac{d}{dx} \sin{x})+(\frac{d}{dx} 7)\]

3
Use Trigonometric Differentiation: the derivative of \(\sin{x}\) is \(\cos{x}\).
\[3\cos{x}+(\frac{d}{dx} 7)\]

4
Use this rule: \(\frac{d}{dx} c=0\).
\[3\cos{x}\]

Done


 

Example 2 [Top]

\[\frac{d}{dx} \sec^{2}x\cos{x}\]
1
Use Product Rule to find the derivative of \(\sec^{2}x\cos{x}\)
The product rule states that \((fg)'=f'g+fg'\).
\[(\frac{d}{dx} \sec^{2}x)\cos{x}+\sec^{2}x(\frac{d}{dx} \cos{x})\]

2
Use Chain Rule on \(\frac{d}{dx} \sec^{2}x\). Let \(u=\sec{x}\).
Use Power Rule: \(\frac{d}{du} {u}^{n}=n{u}^{n-1}\).
\[2\sec{x}(\frac{d}{dx} \sec{x})\cos{x}+\sec^{2}x(\frac{d}{dx} \cos{x})\]

3
Use Trigonometric Differentiation: the derivative of \(\sec{x}\) is \(\sec{x}\tan{x}\).
\[2\sec^{2}x\tan{x}\cos{x}+\sec^{2}x(\frac{d}{dx} \cos{x})\]

4
Use Trigonometric Differentiation: the derivative of \(\cos{x}\) is \(-\sin{x}\).
\[2\sec^{2}x\tan{x}\cos{x}-\sec^{2}x\sin{x}\]

Done


 

Example 3 [Top]

\[\frac{d}{dx} \frac{\tan{x}}{4}+\sin{x}\]
1
Use Sum Rule: \(\frac{d}{dx} f(x)+g(x)=(\frac{d}{dx} f(x))+(\frac{d}{dx} g(x))\).
\[(\frac{d}{dx} \frac{\tan{x}}{4})+(\frac{d}{dx} \sin{x})\]

2
Use Constant Factor Rule: \(\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x))\).
\[\frac{1}{4}(\frac{d}{dx} \tan{x})+(\frac{d}{dx} \sin{x})\]

3
Use Trigonometric Differentiation: the derivative of \(\tan{x}\) is \(\sec^{2}x\).
\[\frac{\sec^{2}x}{4}+(\frac{d}{dx} \sin{x})\]

4
Use Trigonometric Differentiation: the derivative of \(\sin{x}\) is \(\cos{x}\).
\[\frac{\sec^{2}x}{4}+\cos{x}\]

Done