Differentiation & Derivatives Calculator

Differentiation Guide

The Power Rule

For f(x) = axⁿ: f'(x) = anx^(n-1). Examples: d/dx(3x⁴) = 12x³. d/dx(5x) = 5. d/dx(7) = 0 (constants differentiate to zero). d/dx(x^(1/2)) = (1/2)x^(-1/2) = 1/(2√x). d/dx(1/x) = d/dx(x^(-1)) = -x^(-2) = -1/x². Polynomials: differentiate each term separately. d/dx(3x⁴ − 2x² + 5x − 7) = 12x³ − 4x + 5. Chain rule: d/dx(f(g(x))) = f'(g(x)) × g'(x). Example: d/dx((3x+1)⁵) = 5(3x+1)⁴ × 3 = 15(3x+1)⁴.

Standard Derivatives

Trigonometric: d/dx(sin x) = cos x. d/dx(cos x) = −sin x. d/dx(tan x) = sec²x. With chain rule: d/dx(sin(ax)) = a cos(ax). d/dx(cos(ax)) = −a sin(ax). Exponential: d/dx(eˣ) = eˣ (unique — its own derivative). d/dx(e^(ax)) = a e^(ax). d/dx(aˣ) = aˣ ln a. Logarithmic: d/dx(ln x) = 1/x. d/dx(ln(ax)) = 1/x. d/dx(log_a(x)) = 1/(x ln a). These standard results are given in examination data sheets — the key is knowing which rule to apply.

Finding Stationary Points

Stationary points: where dy/dx = 0. To find them: differentiate and set equal to zero. Solve for x. Substitute back to find y coordinates. Nature: use second derivative test. If d²y/dx² > 0: minimum. If d²y/dx² < 0: maximum. If d²y/dx² = 0: inconclusive (use first derivative test — check sign of dy/dx either side of stationary point). Example: y = x³ − 6x² + 9x. dy/dx = 3x² − 12x + 9 = 3(x−1)(x−3) = 0. x = 1 or x = 3. d²y/dx² = 6x − 12. At x=1: −6 < 0 → maximum. At x=3: 6 > 0 → minimum.

Product and Quotient Rules

Product rule: d/dx(uv) = u'v + uv'. Example: d/dx(x² sin x) = 2x sin x + x² cos x. Quotient rule: d/dx(u/v) = (u'v − uv') / v². Example: d/dx(sin x / x) = (cos x · x − sin x · 1) / x² = (x cos x − sin x) / x². Implicit differentiation: for equations of the form f(x,y) = 0. Differentiate both sides with respect to x. Any y terms gain a factor dy/dx. Useful for finding gradients of circles and other implicitly defined curves.