Proving the Derivative of sin(x) is cos(x) using the Limit Definition of Derivative

sin(x)’ =

The derivative of sin(x) is cos(x)

The derivative of sin(x) is cos(x).

To prove this, we can utilize the limit definition of derivative. The derivative of a function f(x) at a point x=a is given by the following limit:

f'(a) = lim(h->0) [f(a+h) – f(a)] / h

Let’s apply this definition to sin(x):

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

Using the trigonometric angle addition identity, we can rewrite the numerator as follows:

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

Now, applying the limit definition:

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

Distributing sin(x) out of the numerator:

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

Now, let’s focus on each term in the numerator separately. Remembering the limit as h approaches zero, we can simplify them as follows:

lim(h->0) [cos(h) – 1] / h = 0

lim(h->0) sin(h) / h = 1

Substituting these simplifications back into the equation:

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

Thus, the derivative of sin(x) is cos(x).

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