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