Q.Differentiate "sin x" by ab-initio method.
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Start your 14-day free trial to unlock the full solution →Using the first-principles (ab-initio) definition of a derivative, d/dx(sin x) works out to cos x.
Let y = sin x. By the ab-initio (first principles) method, the derivative is defined as:
dy/dx = limit as h tends to 0 of [f(x+h) - f(x)] / h
Here f(x) = sin x, so f(x+h) = sin(x+h). Thus:
dy/dx = limit as h tends to 0 of [sin(x+h) - sin x] / h
Using the trigonometric identity sin(C) - sin(D) = 2 cos((C+D)/2) sin((C-D)/2), with C = x+h and D = x:
sin(x+h) - sin(x) = 2 cos(x + h/2) sin(h/2)
Substituting:
dy/dx = limit as h tends to 0 of [2 cos(x + h/2) sin(h/2)] / h
= limit as h tends to 0 of cos(x + h/2) * [sin(h/2) / (h/2)]
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