Mastering Calculus: Discovering The Derivative Of Sin(X) = Cos(X)

d/dx sinx

cosx

The derivative of sin(x) with respect to x is cos(x).

To see why, we can use the definition of the derivative. The derivative of sin(x) is defined as the limit of the difference quotient

(f(x+h) – f(x)) / h

as h approaches 0. Plugging in sin(x) for f(x), we get:

(sin(x+h) – sin(x)) / h

Using the sum-to-product identity for sine, we can simplify this expression to:

[2cos((x+h)/2)sin(h/2)] / h

As h approaches 0, the numerator approaches 2cos(x) and the denominator approaches 1, so the limit of the difference quotient is:

2cos(x) / 1

or simply cos(x). Therefore, the derivative of sin(x) is cos(x).

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