Understanding the Limit Definition of the Derivative
The derivative of a function f(x) at a point x is defined as the limit: f'(x) = lim_{h→0} [f(x+h) - f(x)] / h. For f(x) = sin(x), this becomes sin'(x) = lim_{h→0} [sin(x+h) - sin(x)] / h. To evaluate this, apply the sine angle addition formula: sin(x+h) = sin(x)cos(h) + cos(x)sin(h). Substituting gives [sin(x)cos(h) + cos(x)sin(h) - sin(x)] / h = sin(x)[cos(h) - 1]/h + cos(x)sin(h)/h.
Key Trigonometric Limits
The expression simplifies using standard limits: lim_{h→0} [cos(h) - 1]/h = 0 and lim_{h→0} sin(h)/h = 1. Thus, sin'(x) = sin(x) * 0 + cos(x) * 1 = cos(x). This rigorous proof relies on the squeeze theorem for sin(h)/h and geometric arguments for the cosine limit, confirming the derivative of sin(x) is cos(x).
Practical Example: Deriving at x = π/2
Consider x = π/2. Using the limit: lim_{h→0} [sin(π/2 + h) - sin(π/2)] / h = lim_{h→0} [cos(h) - 1] / h. This matches the cosine limit form, yielding 0, which is cos(π/2). For numerical verification, as h approaches 0 (e.g., h=0.001), [cos(0.001) - 1]/0.001 ≈ -0.0005, converging to 0.
Importance in Calculus and Applications
This derivation is foundational in calculus, establishing trigonometric differentiation rules used in physics (e.g., simple harmonic motion) and engineering (e.g., signal processing). It underscores the power of limits in proving fundamental theorems, avoiding memorization and enabling extensions to other functions like tan(x).