Differentiation
Differentiation is the calculus domain that tests whether you can find instantaneous rates of change from algebraic, trigonometric, exponential, logarithmic, inverse and parametric forms. The board expects correct notation, chain rule fluency and clean simplification.
Core rules
- (x^n)' = n x^(n-1); (sin x)' = cos x; (cos x)' = -sin x; (e^x)' = e^x; (ln x)' = 1/x.
- Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v - uv')/v^2.
- Chain rule: dy/dx = dy/du * du/dx. Every inner function must be differentiated.
- Implicit differentiation: differentiate both sides with respect to x, then solve for dy/dx.
- Parametric: if x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.
- Logarithmic differentiation suits y = [f(x)]^{g(x)}: take ln y first.
Worked example
Differentiate y = x^x, x > 0. Take natural logs: ln y = x ln x. Differentiate implicitly: (1/y)(dy/dx) = ln x + 1. Hence dy/dx = x^x(ln x + 1). The power rule alone is invalid because both base and exponent vary.
Mark-losing traps
- Forgetting the inner derivative in sin(3x^2), ln(cos x) or e^(x^2).
- Applying the power rule to x^x or a^x without adjusting.
- Omitting dy/dx during implicit differentiation.
- Using dx/dt divided by dy/dt for parametric slopes.
- Leaving a derivative in mixed variables instead of simplifying.