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