Question

find the derivative of cos X from first principle​

Answer 1

HOPE YOU GOT YOUR ANSWER FROM THIS.

Answer 2

derivative of cos X is -sin x

Step-by-step explanation:

here ,

f(x) =cos x

we have to find the f'(x)

then

f'(x) = $\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

f(x) = cos x

f(x+h) = cos (x+h)

putting the value we get

f'(x) = $\lim_{h\rightarrow 0}\frac{cos (x+h)-cos(x)}{h}$

using the formula

cos A - cos B = $- 2 sin(\frac{A+B}{2}) sin (\frac{A-B}{2})$

f'(x) = $\lim_{h\rightarrow 0}\frac{-2 sin (\frac{x+x+h}{2}). sin(\frac{x+h-x}{2}) }{h}$

= $\lim_{h\rightarrow 0}\frac{-2 sin (\frac{2x+h}{2}). sin(\frac{h}{2}) }{h}$

on calculating we get,

$\lim_{h\rightarrow 0}{- sin(\frac{2x+h}{2}).1}$

$\lim_{h\rightarrow 0}{- sin(\frac{2x+h}{2})}$

h = 0

= $\lim_{h\rightarrow 0}{- sin(\frac{2x+0}{2})}$

= $\lim_{h\rightarrow 0}{- sin(\frac{2x}{2})}$

= - sin x

hence ,

derivative of cos X is -sin x

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