Derivative of Trigonometric Functions

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

Example 1

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

Example 2

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

Example 3

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

Example 4

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

Example 5

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

Example 6

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

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 $\dfrac{d}{dx}e^x = e^x$.

Example 8

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