Ddxsinx

Ddxsinx is a mathematical function that represents the derivative of the sine function with respect to a variable. The function is a fundamental concept in calculus and is denoted as the derivative of sin(x).

The derivative of sin(x), or d/dx sin(x), is given by d/dx sin(x) = cos(x). This means that the rate of change of the sine function with respect to x is equal to the cosine function.

To derive this expression, we can use the limit definition of a derivative. The derivative of a function f(x) is defined as f'(x) = lim(h -> 0) [f(x + h) - f(x)]/h. Applying this definition to the sine function gives d/dx 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 simplify the expression to sin(x + h) - sin(x) = -2sin((x + h/2))sin(h/2). Taking the limit as h approaches 0 gives d/dx sin(x) = cos(x).

The derivative of the sine function is an important concept in calculus and has numerous applications in physics, engineering, and other fields. It is used to describe the rate of change of the sine function and is essential for solving optimization problems and modeling real-world phenomena.