Differentiation techniques — A/L Combined Mathematics

What must I remember about Differentiation techniques?

• Derivative from first principles: f'(x) = lim_h->0 [f(x + h) - f(x)] / h; represents instantaneous rate of change of f with respect to x, and gradient of tangent to y = f(x).
• Standard derivatives: d/dx(xⁿ) = n*x^n-1; d/dx(e^x) = e^x; d/dx(ln x) = 1/x; d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(tan x) = sec² x; d/dx(sec x) = sec x tan x; d/dx(cosec x) = -cosec x cot x; d/dx(cot x) = -cosec² x.
• Differentiation rules: Product Rule: d/dx(uv) = u*(dv/dx) + v*(du/dx); Quotient Rule: d/dx(u/v) = [v*(du/dx) - u*(dv/dx)] / v²; Chain Rule: dy/dx = (dy/du) * (du/dx).
• Implicit differentiation: When y cannot be explicitly isolated (e.g., x² + y² = r²), differentiate term by term with respect to x treating y as y(x), applying d/dx(yⁿ) = n*y^n-1*(dy/dx).
• Parametric differentiation: If x = f(t) and y = g(t), then dy/dx = (dy/dt) / (dx/dt); second derivative d^2y/dx² = [d/dt(dy/dx)] / (dx/dt).

Test yourself (1 minute)

1. Second derivative of parametric function pitfall

   d^2y/dx² is NOT (d^2y/dt²)/(d^2x/dt²); it MUST be computed as d/dt(dy/dx) divided by dx/dt.

2. Logarithmic differentiation technique

   For functions of the form y = [f(x)]^g(x) or complex products/quotients, take ln on both sides: ln y = g(x) * ln f(x), then differentiate implicitly.

3. Tangents and Normals

   Gradient of tangent is m = (dy/dx)|_{(x1, y1)}; equation of tangent is y - y1 = m(x - x1); gradient of normal is -1/m; equation of normal is y - y1 = (-1/m)(x - x1).

Practise Differentiation techniques questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 14

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Year 1 · Term 2