Calculus Cheat Sheet (Derivatives)

Feynman’s technique is the greatest integration method of all time
Feynman’s technique is the greatest integration method of all time

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

  1. 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…

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