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:

Euler’s notation: Euler’s notation uses a capital DD to indicate the derivative of a function. So in Euler’s notation:

The following are equivalent notations for the derivative function where y = f(x):

If x = f(t) we include Newton’s notation so we have:

Properties of the derivative and methods for finding the derivative function

- Multiplication by a scalar (a consequence of linearity)

where a is some constant and u = u(x) is some function of x.

2. The sum rule (the other consequence of linearity)

where u = u(x), v = v(x), and w = w(x) are all functions of x.

3. The power rule

where u = u(x) is some function of x and n is any real valued number. Note that du/dx is an example of the chain rule. It’s…