Differentiation

The derivative of f(x) at point

The derivative of f(x) at point x is defined as f'(x) = lim(h→0) [f(x+h) - f(x)]/h. This represents the instantaneous rate of change of the function a

The derivative dy/dx at any point

The derivative dy/dx at any point P(x,y) on curve y = f(x) represents the slope of the tangent line at that point. If the tangent makes angle θ with x

For any function y = x^n

For any function y = x^n, the derivative is dy/dx = nx^(n-1). Examples: d/dx(x³) = 3x², d/dx(x^(-1)) = -x^(-2) = -1/x², d/dx(√x) = d/dx(x^(1/2)) = (1/

d/dx[f(x) ± g(x)] = f'(x) ±

d/dx[f(x) ± g(x)] = f'(x) ± g'(x). For example, if y = x³ + 2x² - 5x + 7, then dy/dx = 3x² + 4x - 5. The derivative of a constant is zero: d/dx(c) = 0

d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x). Example: If y = (x² + 1)(3x - 2), then dy/dx = (2x)(3x - 2) + (x² + 1)(3) = 6x² - 4x + 3x² + 3 = 9x² - 4x +