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**Derivatives of Trigonometric Functions:**

Trigonometric and inverse trigonometric functions are differentiable at each point of its domain.

- $latex \frac{d}{dx} \sin x = \cos x $
- $latex \frac{d}{dx} \cos x = – \sin x $
- $latex \frac{d}{dx} \tan x = \sec^2 x $
- $latex \frac{d}{dx} \cot x = – cosec^2 x $
- $latex \frac{d}{dx} \sec x = \sec x . \tan x$
- $latex \frac{d}{dx} cosec x = – cosec x. \cot x$

**Note:** remember that the trigonometric functions which are starts with **c** then their derivate should come with the minus(-) sign preceding the derivative of the functions and also applicable in case of **the inverse trigonometric functions** Ex: $latex \frac{d}{dx} \cos x = – \sin x$

**Derivatives of Inverse Trigonometric Functions:**

- $latex \frac{d}{dx} \sin^{-1}x = \frac{1}{\sqrt{1- x^2}}, (|x|<1)$
- $latex \frac{d}{dx} \cos^{-1}x = – \frac{1}{\sqrt{1- x^2}}, (|x|<1)$
- $latex \frac{d}{dx} \tan^{-1}x = \frac{1}{1+x^2}$, $latex x \in R $
- $latex \frac{d}{dx} \cot^{-1}x = -\frac{1}{1+x^2}$, $latex x \in R $
- $latex \frac{d}{dx} \sec^{-1}x = \frac{1}{x \sqrt{x^2-1}}, (|x|>1)$
- $latex \frac{d}{dx} cosec^{-1}x = -\frac{1}{x \sqrt{x^2- 1}}, (|x|>1)$

**Derivatives of Exponential and Logarithmic Functions:**

- $latex \frac{d}{dx}e^x = e^x$
- $latex \frac{d}{dx}a^x = a^x \log_ea, a>0, a \neq 1$
- $latex \frac{d}{dx}\log_ex = \frac{1}{x}, x>0$
- $latex \frac{d}{dx} \log_ax = \frac{1}{xlog_ea} = \frac{log_ae}{x}$

The logarithmic function is differentiable at each point of its domain. $latex a^x$ is differentiable at each $latex x \in , (a >0, a\neq 1)$

**Derivatives of Hyperbolic Functions:**

- $latex \frac{d}{dx} \sin h x = \cos h x $
- $latex \frac{d}{dx} \cos h x = – \sin h x $
- $latex \frac{d}{dx} \tan h x = \sec h^2 x $
- $latex \frac{d}{dx} \cot h x = – cosec h^2 x $
- $latex \frac{d}{dx} \sec h x = \sec h x . \tan h x$
- $latex \frac{d}{dx} cosec h x = – cosec h x. \cot h x$

**Parametric Differentiation**

If x = f(t), y =g(t) are both the derivable functions of ‘t’ then

$latex \frac{d}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}, (f'(t) \neq 0)$

**Derivative of Infinite series:**

- If $latex y = \sqrt{f(x) + \sqrt{f(x) + \sqrt{f(x)+ \cdots + \infty}}}$ then $latex y = \sqrt{f(x)+y}$

Squaring:

$latex y^2 = f(x) +y \\y^2-y = f(x)$

Differentiating both sides w.r.t. x, we get

$latex (2y-1) \frac{dy}{dx} = f'(x)$

For example let take one trigonometric function:

$latex y = \sqrt{\sin x+ \sqrt{ \sin x + \sqrt{\sin x+ \cdots + \infty}}}$ then $latex (2y -1) \frac{dy}{dx} = \cos x$

**Note: **do like this for each and every functions to get derivative of the functions.