Trigonometric functions

$$ 1. \ \frac {d}{dx} \sin x= \cos x$$

$$ 2. \ \frac {d}{dx} \cos x = - \sin x $$

$$ 3. \ \frac {d}{dx} \tan x = \sec^2 x $$

$$ 4. \ \frac {d}{dx} \cot x = - cosec^2 x $$

$$ 5. \ \frac {d}{dx} \sec x = \sec x \cdot \tan x $$

$$ 6. \ \frac {d}{dx} cosec \ x = - cosec \ x \cdot \cot x $$

Example 1:

$$ Differentiate \ y = \sin (2x+5) $$

Solution:

$$ \frac {dy}{dx} = \frac {d}{dx} \sin (2x+5)$$

$$ \frac {dy}{dx} = \cos (2x+5) \frac {d}{dx} \sin (2x+5) $$

$$ \frac {dy}{dx} = \cos (2x+5) [2] $$

$$ \frac {dy}{dx} = 2 \cos (2x+5) $$

Example 2:

$$ Differentiate \ y= \tan^2x $$

Solution:

$$ \frac {dy}{dx} = \frac {d}{dx} \tan^2x $$

$$ \frac {dy}{dx} = 2 \tan^{2-1}x \cdot \frac {d}{dx} \tan x $$

$$ \frac {dy}{dx} = 2 \tan x \cdot \sec^2x $$

Example 3:

$$ Differentiate \ y = \cot (4 – 3x). $$

Solution:

$$ \frac {dy}{dx} = -cosec^2(4-3x)(-3) $$

$$ \frac {dy}{dx} = 3cosec^2(4-3x) $$