Differentiation

When differentiating y = f(g(x))

When differentiating y = f(g(x)), use dy/dx = (dy/du) × (du/dx) where u = g(x). For example, if y = (4x³ + 3x² - 2x)⁶, let u = 4x³ + 3x² - 2x, then y

If y = f(x) has

If y = f(x) has an inverse x = f⁻¹(y), then dx/dy = 1/(dy/dx), provided dy/dx ≠ 0. This is crucial in economics for finding marginal demand. For examp

For functions involving products

For functions involving products, quotients, or variable exponents like x^x, take the natural logarithm of both sides first. For y = x^x: Step 1: ln y

When variables cannot be separated (like

When variables cannot be separated (like x² + y² = a²), differentiate both sides with respect to x, treating y as a function of x. For x² + y² = a²: S

When both x and y

When both x and y are functions of a parameter t (like x = f(t), y = g(t)), use dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. For example, if x = 2at a