Calculus Cheat Sheet (Derivatives)

Definition and Notation

If y = f(x) and f(x) is smooth and continuous (differentiable) at x then:

• m = f ′(a) is the slope of the line tangent to f(a) if f(x) is differentiable at a.
• The equation of the line tangent to f(x) at a is given by:
• f ′(a) is the instantaneous rate of change of f(x) at a.

Leibniz’s notation: The most prolifically used notation in mathematics for the derivative is the Leibniz notation. The numerator and denominator are sometimes individually referred to as infinitesimals. Their ratio, i.e. dy/dx is often referred to as the differential. The notation as follows:

simply means the infinitesimal change in y given an infinitesimal change in x or the infinitesimal change in the value of f(x) given an infinitesimal change in x. Second and third derivatives are written thus:

Lagrange’s notation: Another commonly used notation is the Lagrange notation. This notation uses the prime mark to indicate the derivative of a function. So if f is a function of x then:

Newton’s notation: Newton’s notation is most often used in physics where the independent variable is time. In Newton’s notation where x is a function of t first, second, and third derivatives are written as: