Proving the Derivative of sin(x) = cos(x) with the Fundamental Formula

Derivative of sin x

cos x

The derivative of sin x is cos x.

To prove this, we can use the fundamental derivative formula that states that the derivative of a function f(x) with respect to x equals the limit of (f(x+h)-f(x))/h as h approaches zero.

So, let’s apply this formula to the function f(x) = sin x:

f'(x) = lim (sin(x+h) – sin(x))/h as h approaches zero

We know that sin(x+h) can be written as sin x cos h + cos x sin h (using trigonometric identities).

So, f'(x) = lim (sin(x)cos(h) + cos(x)sin(h)-sin(x))/h as h approaches zero

Rearranging the terms, we get

f'(x) = lim [(sin(x)cos(h)-sin(x))/h] + lim[cos(x)sin(h)/h] as h approaches zero

The first limit evaluates to 0 since sin x is a constant and cos h approaches 1 as h approaches zero.

The second limit can be simplified using the definition of the derivative of sin x at x=0 (which is 1).

So, f'(x) = 0 + cos x (lim sin(h)/h as h approaches zero)

The limit sin(h)/h approaches 1 as h approaches zero (which can be proven using L’Hopital’s rule or by using the squeeze theorem).

Thus, f'(x) = cos x

Hence, the derivative of sin x is cos x.

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