# 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 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}$Done3*cos(x)  Example 2 [Top]$\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 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 The derivative of $$\cos{x}$$ is $$-\sin{x}$$$2\sec^{2}x\tan{x}\cos{x}-\sec^{2}x\sin{x}$Done2*sec(x)^2*tan(x)*cos(x)-sec(x)^2*sin(x)  Example 3 [Top]$\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 The derivative of $$\tan{x}$$ is $$\sec^{2}x$$$\frac{\sec^{2}x}{4}+(\frac{d}{dx} \sin{x})$4 The derivative of $$\sin{x}$$ is $$\cos{x}$$$\frac{\sec^{2}x}{4}+\cos{x}$Donesec(x)^2/4+cos(x)