Proving The Derivative Of Sin(X) With Respect To X Is Cos(X) Using Trigonometric Identities And Limit Definition Of Derivative.

d/dx sinx

cosx

The derivative of sin(x) with respect to x is cos(x). Mathematically, we write:

d/dx sin(x) = cos(x)

The proof of this can be done by using the definition of the derivative and the basic trigonometric identities. One way to prove this is by using the limit definition of the derivative:

f'(x) = lim(h->0) [(f(x+h) – f(x))/h]

If we plug in sin(x) for f(x), we get:

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

We then apply the trigonometric identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b) to the numerator:

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

The sin(x) terms cancel out, and we are left with:

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

As h approaches 0, the sin(h)/h term approaches 1, so we can simplify the expression to:

sin'(x) = cos(x)

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

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