Differential Calculus

Differential Calculus studies the rates at which quantities change, I.e. continuous change, or rate of change of a function. it focuses on obtaining the solution to the problems where the rate of a function changes or different properties of integrals and derivatives of functions should be determined.


  • Limit: , and , here is called the limit to the function , which can be written as
  • Interval: An interval defines as the range of numbers that are existing between the two given numbers.
    - Open interval: represent all possible complete real numbers in a particular set represented as where .
    - Closed interval: represent all possible complete real numbers in a particular set represented as where .
    - Half Closed Interval:
  • Domain: The domain of a function is the input values of a function.
  • Range: The range of a function is the output value of a function.


The derivatives of the function determine the rate of change of a function at a point, it is also called the slope of a function and therefore also defined as the ratio of the change in the value of a dependent variable to the change in the independent variable. The procedure used to obtain the derivatives is called differentiation.

in function , where and , derivative is represented as or .

Derivative gives us the slope at the point x. Meaning that it describes how a small change in the input x will affect output y.

If this means that there is no slope, meaning small changes in the will not change the value of . This means we have hit stationary point at point . Minimum and maximum are both types of stationary points. The third type of stationary point is so-called saddle point. It has slope zero but it is no minimum nor maximum.

partial derivative is used in functions with multiple input.

Basic Rules of Differential Calculus

  • The Constant Rule
  • The Power Rule
  • Multiplied by a Constant Rule
  • Sum Rule
  • Difference Rule
  • Product Rule
  • Quotient Rule
  • Sum/Difference involving a Constant Rule

Differential Calculus Formulas

Function Differentiation Formula
C (Constant Function)
(a line)
Square Function
(Power Function)
Exponential Functions ,
Log Functions ,
Trigonometric Functions , , , , ,
Inverse Trigonometric Functions , ,