Understanding the Derivative of the Sine Function | Why it Equals Cosine

Derivative of: sin(x)

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

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

To understand why the derivative of sin(x) is cos(x), we can use the definition of the derivative. The derivative of a function represents the rate of change of the function at any given point. In the case of sin(x), it represents the rate of change of the sine function as the input x changes.

Using the definition of the derivative, we have:

f'(x) = lim(h->0) [f(x+h) – f(x)] / h

Applying this to the sine function, we get:

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

Using the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite the numerator:

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

Applying the limit and simplifying, we get:

sin'(x) = cos(x)

Therefore, the derivative of sin(x) is cos(x). This means that for any value of x, the rate of change of the sine function at that point is equal to the cosine of the same point.

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