Differentiation – A Level Mathematics CAIE Revision

Fundamental Concepts of Differentiation

Differentiation: Process used to find the rate of change of a quantity. It provides the derivative of a function.
Derivative: Outcome of differentiation, representing the gradient of a curve at any given point.
First and Second derivatives: Differentiating once gives the first derivative, usually represented as f’(x) or dy/dx. Differentiating the derivative gives the second derivative, usually represented as f’‘(x) or d²y/dx².

Power rule: If y = x^n, then dy/dx = n*x^(n-1).
Sum/Difference rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives.
Product rule: If y = u*v, then dy/dx = u’v + uv’.
Quotient rule: If y = u/v, then dy/dx = (vu’ - uv’)/ (v)^2.
Chain rule: Used for differentiating a function of a function, i.e., if y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).

Tangents and Normals: The derivative at a point gives the slope of the tangent or the normal at that point on the curve.
Maxima and Minima: The points at which f’(x)=0 represent local extrema; f’‘(x) is used to determine whether the extrema are maxima, minima or points of inflection.
Rates of change: Differentiation provides a method to determine how variables change with respect to each other.
Small increments: The derivative is used in approximations of small changes in the function, aiding in understanding movements around a specific x-value.

Implicit differentiation: A method for finding derivatives when it’s difficult or impossible to express one variable explicitly in terms of the other(s).
Logarithmic differentiation: A method used to differentiate functions that may be difficult to differentiate normally.
Parametric differentiation: A method used when a function is given as a system of equations (parametric equations), providing dy/dx and d²y/dx² in terms of a parameter.

Partial derivatives: If a function involves more than one variable, each variable can be differentiated while treating the other variables as constant.
Higher order derivatives: The process of finding derivatives can be applied multiple times, resulting in second-order, third-order derivatives, and so forth.