Differential Calculus
Core concepts
Differentiation measures instantaneous rate of change. You must know the limit definition of the derivative, the power, product, quotient, and chain rules, and derivatives of exponential, logarithmic, and trigonometric functions. Interpretations matter: gradient of a tangent, velocity from displacement, and optimisation problems where the derivative is set to zero.
The derivative is zero at stationary points; the second derivative distinguishes maxima (negative), minima (positive), and possible points of inflection (zero). Increasing behaviour corresponds to a positive derivative, decreasing to a negative one. First-principles differentiation occasionally appears and requires careful algebra with limits.
Traps to watch
- Forgetting the chain rule when differentiating composite functions such as sin(3x²).
- Applying the product rule incorrectly, e.g. differentiating a product as the product of derivatives.
- Assuming every point where f′(x) = 0 is a maximum or minimum; it may be a point of inflection.
- Losing the negative sign when differentiating cos x or differentiating ln x incorrectly.
Rule of thumb
Identify the outermost structure first: is it a product, quotient, or composition? Apply the corresponding rule, then simplify before substituting.
Worked example
Differentiate y = x²e^(3x). This is a product, so y′ = 2x·e^(3x) + x²·3e^(3x) = e^(3x)(2x + 3x²). Setting y′ = 0 gives x = 0 or x = −⅔; the second derivative test confirms which is a maximum and which is a minimum.