# Derivative of Trigonometric Functions

\large \displaystyle \begin{align} \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \dfrac{d}{dx}\cos{x} &= -\sin{x} \\ \dfrac{d}{dx}\tan{x} &= \sec^2{x} \\ \end{align}

### Example 1

Prove $\dfrac{d}{dx}\tan{x} = \sec^2{x}$ using $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}\cos{x} = -\sin{x}$.

\begin{align} \displaystyle \require{AMSsymbols} \require{AMSsymbols} \require{color} \dfrac{d}{dx}\tan{x} &= \dfrac{d}{dx}\dfrac{\sin{x}}{\cos{x}} \\ &= \dfrac{\dfrac{d}{dx}\sin{x} \times \cos{x}-\sin{x} \times \dfrac{d}{dx}\cos{x}}{\cos^2{x}} &\color{red} \text{quotient rule}\\ &= \dfrac{\cos{x} \times \cos{x}-\sin{x} \times (-\sin{x})}{\cos^2{x}} \\ &= \dfrac{\cos^2{x} + \sin^2{x}}{\cos^2{x}} \\ &= \dfrac{1}{\cos^2{x}} &\color{red} \cos^2{x} + \sin^2{x}=1\\ &= \sec^2{x} \end{align}

### Example 2

Find $\dfrac{dy}{dx}$ for $y=\sin{(2x)}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\sin{(2x)} \\ &= \cos{(2x)} \times \dfrac{d}{dx}2x \\ &= \cos{(2x)} \times 2 \\ &= 2\cos{(2x)} \end{align}

### Example 3

Find $\dfrac{dy}{dx}$ for $y=\cos{(x^2+1)}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\cos{(x^2+1)} \\ &= -\sin{(x^2+1)} \times \dfrac{d}{dx}(x^2+1) \\ &= -\sin{(x^2+1)} \times 2x \\ &= -2x\sin{(x^2+1)} \end{align}

### Example 4

Find $\dfrac{dy}{dx}$ for $y=\tan{x^4}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\tan{x^4} \\ &= \sec{x^4} \times \dfrac{d}{dx}x^4 \\ &= \sec{x^4} \times 4x^{4-1} \\ &= 4x^{3}\sec{x^4} \\ \end{align}

### Example 5

Find $\dfrac{dy}{dx}$ for $y=\sin{2x}\cos{5x}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\sin{2x}\cos{5x} \\ &= \dfrac{d}{dx}\sin{2x} \times \cos{5x} + \sin{2x} \times \dfrac{d}{dx}\cos{5x} \\ &= \cos{2x} \times \dfrac{d}{dx}2x \times \cos{5x} + \sin{2x} \times (-\sin{5x}) \times \dfrac{d}{dx}5x \\ &= \cos{2x} \times 2 \times \cos{5x} + \sin{2x} \times (-\sin{5x}) \times 5 \\ &= 2\cos{2x}\cos{5x}-5\sin{2x}\sin{5x} \end{align}

### Example 6

Find $\dfrac{dy}{dx}$ for $y=x\tan{x}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}x\tan{x} \\ &= \dfrac{d}{dx}x \times \tan{x} + x \times \dfrac{d}{dx}\tan{x} \\ &= 1 \times \tan{x} + x \times \sec^2{x} \\ &= \tan{x} + x\sec^2{x} \end{align}

## Extension Examples

These Extension Examples require to have some prerequisite skills, including;
\begin{align} \displaystyle \dfrac{d}{dx}e^x &= e^x \\ \dfrac{d}{dx}\log_e{x} &= \dfrac{1}{x} \\ \end{align}

### Example 7

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\dfrac{e^{2x}}{\sin{(3x)}}$, known that $\dfrac{d}{dx}e^x = e^x$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\dfrac{e^{2x}}{\sin{(3x)}} \\ &= \dfrac{\dfrac{d}{dx}e^{2x} \times \sin{(3x)}-e^{2x} \times \dfrac{d}{dx}\sin{(3x)}}{\sin^2{(3x)}} \\ &= \dfrac{2e^{2x} \times \sin{(3x)}-e^{2x} \times 3\cos{(3x)}}{\sin^2{(3x)}} \\ &= \dfrac{2e^{2x}\sin{(3x)}-3e^{2x}\cos{(3x)}}{\sin^2{(3x)}} \end{align}

### Example 8

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\log_e{\tan{x^2}}$, known that $\dfrac{d}{dx}\log_e{x} = \dfrac{1}{x}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\log_e{\tan{x^2}} \\ &= \dfrac{1}{\tan{x^2}} \times \dfrac{d}{dx} \tan{x^2} \\ &= \dfrac{1}{\tan{x^2}} \times \sec^2{x^2} \times \dfrac{d}{dx}x^2 \\ &= \dfrac{1}{\tan{x^2}} \times \sec^2{x^2} \times 2x \\ &= \dfrac{2x}{\tan{x^2}}\sec^2{x^2} \end{align}

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