How do I differentiate sin(x) from first principles?

Question

Our teacher says we should be able to derive the derivative of sin x using the definition, not just memorise it. I get stuck after writing the difference quotient.

Deepak Kulkarni · Accepted Answer

Start from the definition f'(x) = lim(h→0) [f(x+h) − f(x)]/h. With f(x) = sin x, you get [sin(x+h) − sin x]/h. Expand sin(x+h) = sin x cos h + cos x sin h. The numerator becomes sin x(cos h − 1) + cos x sin h. Split into two fractions: sin x·(cos h − 1)/h + cos x·(sin h)/h. Now use the two standard limits: as h→0, (sin h)/h → 1 and (cos h − 1)/h → 0. So the first term vanishes and the second gives cos x·1. Therefore the derivative of sin x is cos x. The same method gives the derivative of cos x as −sin x.

How do I differentiate sin(x) from first principles?

1 answer