Differentiation finds the rate of change of a function. It underpins optimisation, curve sketching, kinematics, and many applications across mathematics and science.
Power rule: d/dx(xⁿ) = nxⁿ⁻¹. Constant multiplier: d/dx(kf(x)) = kf\'(x). Sum rule: d/dx(f+g) = f\'+g\'. Standard results: d/dx(sinx) = cosx, d/dx(cosx) = -sinx, d/dx(eˣ) = eˣ, d/dx(lnx) = 1/x. Chain rule: d/dx(f(g(x))) = f\'(g(x)) · g\'(x) — "differentiate the outside, multiply by derivative of inside".
Product rule: d/dx(uv) = u(dv/dx) + v(du/dx). Quotient rule: d/dx(u/v) = (v(du/dx) - u(dv/dx))/v². Use product rule when two functions are multiplied, quotient rule when divided. Example: d/dx(x²sinx) = x²cosx + 2xsinx (product rule).
Stationary points: dy/dx = 0. Nature: d²y/dx² > 0 → minimum, d²y/dx² < 0 → maximum, d²y/dx² = 0 → test fails (check gradient either side). Gradient of tangent at point: evaluate dy/dx at that point. Normal: perpendicular to tangent. Connected rates of change: chain rule links related rates, e.g., dV/dt = dV/dr × dr/dt. Small increments: δy ≈ (dy/dx)δx gives approximate change.
Use the chain rule when you have a function of a function (composite): e.g., (2x+1)⁵, sin(3x), e^(x²) — there is an "inner" and "outer" function. Use the product rule when two separate functions are multiplied together: e.g., x²sinx, xe^x, (x+1)lnx. Use the quotient rule when one function is divided by another: e.g., sinx/x, x²/(x+1). Often you need to combine rules: e.g., x²sin(3x) needs the product rule for x² × sin(3x), and the chain rule when differentiating sin(3x).
Book a Trial + Diagnostic session. Get a personalized Learning Path with clear milestones, tutor match, and a plan recommendation — all within 24 hours.
Book Trial + Diagnostic →