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प्रश्न
Find the derivative of the following function.
sin x cos x
उत्तर
Let f (x) = sin x cos x. Accordingly, from the first principle,
f'(x) = `lim_(h->0) (f(x + h) - f(x))/h`
= `lim_(h->0) (sin (x + h) cos(x + h) - sin x cos x)/h`
= `lim_(h->0)1/(2h)[2sin(x + h) cos(x + h) -2 sin x cos x]`
= `lim_(h->0)1/(2h)[sin2 (x + h)-sin2x]`
= `lim_(h-_0)1/(2h)[2cos (2x + 2h + 2x)/2 . sin (2x + 2h -2x)/2]`
= `lim_(h->0)1/h [cos (4x + 2h)/2 sin (2h)/h]`
= `lim_(h->0)1/h [cos (2x + h) sin h]`
= `lim_(h->0) cos (2x + h). lim_(h->0)sin h/h]`
= cos (2x + 0).1
= cos 2x
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