Derivative of Sine Function | Explained with Step-by-Step Calculations and Derivation

Derivative of sinx

The derivative of the sine function (sin(x)) can be found using basic differentiation rules

The derivative of the sine function (sin(x)) can be found using basic differentiation rules.

Let’s use the definition of the derivative:

f'(x) = lim(h->0) [f(x + h) – f(x)] / h

Applying this to the sine function:

f(x) = sin(x)
f(x + h) = sin(x + h)

Using the angle addition identity for sine:

sin(x + h) = sin(x)cos(h) + cos(x)sin(h)

Now, substitute these into the definition of the derivative:

f'(x) = lim(h->0) [(sin(x)cos(h) + cos(x)sin(h)) – sin(x)] / h

Simplifying:

f'(x) = lim(h->0) [sin(x)cos(h) + cos(x)sin(h) – sin(x)] / h

Now, let’s collect the terms with h:

f'(x) = lim(h->0) [sin(x)cos(h) + cos(x)sin(h) – sin(x)] / h
= lim(h->0) [sin(x)cos(h)/h + cos(x)sin(h)/h – sin(x)/h]

As h approaches zero, sin(h)/h approaches 1 (this can be rigorously proved using limits).

Therefore, we have:

f'(x) = lim(h->0) [sin(x)cos(h)/h + cos(x)sin(h)/h – sin(x)/h]
= sin(x)lim(h->0) [cos(h)/h] + cos(x)lim(h->0) [sin(h)/h] – lim(h->0) [sin(x)/h]
= sin(x)(0) + cos(x)(1) – (0)
= cos(x)

So, the derivative of sin(x) with respect to x is cos(x).

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