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प्रश्न
Find the derivative of cos x from first principle.
उत्तर
Let f(x) = cos x
∴ f(x + h) = cos (x + h)
f(x + h) − f(x) = cos (x + h) − cos x
= `lim_(h->0)[(-2 sin ((x + h + x)/2) sin ((x + h - x)/2))/h]`
= `lim_(h->0) [(-2sin((2x + h)/2) sin(h/2))/h]`
= `lim_(h->0)[(-2 sin (x + h/2) sin (h/2))/h]`
∴ f'(x) = `lim_(h → 0) (f(x + h) - f(x))/h`
= `lim_(h → 0)(-2sin (x + h/2) sin h/2)/h`
= `lim_(h → 0)[-sin(x+ h/2)]((sin h/2)/(h/2))`
= `- sinx` ...........`[∵ lim_(h → 0) (sin h/2)/(h/2) = 1]`
Hence, `d/dx cos x = - sin x`
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