rules differentiation

Differentiation is the process of finding the rate of change of a function with respect to its variable. The derivative of a function f(x), denoted f'(x) or dy/dx, represents the gradient of the tangent to the curve at any point.

The Power Rule is fundamental: If f(x) = axⁿ, then f'(x) = naxⁿ⁻¹. This works for all real values of n, including negative and fractional powers. Remember the mnemonic: "Bring down the power, reduce by one".

The Sum/Difference Rule states that derivatives of sums are sums of derivatives: d/dx[f(x) ± g(x)] = f'(x) ± g'(x). You differentiate each term independently.

The Product Rule applies when multiplying functions: If y = uv, then dy/dx = u(dv/dx) + v(du/dx). Mnemonic: "First times derivative of second, plus second times derivative of first" or "u'v + uv'".

The Quotient Rule handles division: If y = u/v, then dy/dx = [v(du/dx) - u(dv/dx)]/v². Mnemonic: "Bottom times derivative of top, minus top times derivative of bottom, all over bottom squared" or "(vu' - uv')/v²".

The Chain Rule differentiates composite functions: If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). Think "derivative of outside times derivative of inside". Alternative notation: dy/dx = (dy/du) × (du/dx).

Cambridge Standard: Always show clear working when applying these rules. The mark scheme rewards systematic application of formulae.

Understanding why these rules work illuminates their power. The power rule emerges from first principles: as x changes by δx, xⁿ changes proportionally to nxⁿ⁻¹δx when δx approaches zero.

Real-World Application: Physics & Motion If a particle's position is s(t) = 5t³ - 2t² + 7t metres after t seconds, the velocity v(t) = s'(t) = 15t² - 4t + 7 m/s, and acceleration a(t) = v'(t) = 30t - 4 m/s². Engineers use these derivatives to design braking systems and optimize trajectories.

Economics: Marginal Analysis If a company's profit function is P(x) = -2x² + 100x - 500 (where x is units produced), then P'(x) = -4x + 100 represents marginal profit – the additional profit from producing one more unit. When P'(x) = 0, profit is maximized.

The Chain Rule Analogy Imagine you're converting temperatures: Celsius to Kelvin to Rankine. The rate of change of Rankine with respect to Celsius requires multiplying the rates at each stage – exactly like the chain rule multiplies derivatives through composite functions.

Product Rule Intuition When two quantities both change (like length and width of an expanding rectangle), the area's rate of change includes contributions from both dimensions changing: d/dt(length × width) = (rate of length change × width) + (length × rate of width change).

Key Insight: These rules aren't arbitrary – they reflect how interconnected quantities genuinely behave in nature and commerce.