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  • Basic laws of differentiation
    Let f (x) and g(x) be any two functions of x. Then,
  • Scalar multiple rule
    Description: 88071.png
    where c is any constant
  • Sum and difference rule
    Description: 88067.png
  • Product rule
    Description: 88063.png 
  • Quotient rule
    Description: 88059.png 
  • Derivative of a function of a function
    Let y = f (u), where u = g(x)
    ThenDescription: 88055.png 
  • Implicit functions
    A function f (xy) = 0 is said to be an implicit function, if y cannot be directly defined as a function of x.
    In such a case, we will differentiate both sides of this equation w.r.t x, collect the terms containing Description: 88051.png on one side; transfer other terms to the other side and divide by the coefficient of  Description: 88047.png to get its value.
  • Parametric equation
    If x = f (t) and y = g(t), then Description: 88043.png
  • Logarithmic differentiation
    When a function is expressed in any of the following forms, its derivative can be obtained by taking the logarithm of the function and then differentiating it.
    (i) A product of a number of functions
    (ii) When a function is raised to some exponent which is also a function
    (iii) A number of functions are divided
    This method of differentiation is known as logarithmic differentiation.
  • Higher order derivatives
    Let y = f (x), be a function of xDescription: 88039.png is called the first derivative of y with respect to x.
    The derivative of f ′(x) is called the second derivative of y with respect to x,
    i.e.Description: 88035.png
  • Integral Calculus
    Integration is the inverse operation of differentiation.
    If the derivative of y with respect to x is given by Description: 88029.png, the integral of f ′(x) with respect to x is given byDescription: 88025.png
  • Basic laws of integration
    Let f (x) and g(x) be any two functions of x, then
    Description: 88021.png
    Description: 88017.png 

Integration by Substitution

Sometimes it is not possible to be able to integrate f (x) directly. We can substitute f (x) into some other function g (t) to make it readily integrable.

Description: 88013.png 

Description: 88009.png 

Integration by Parts

Let u = f (x) and v = g(x) be two different functions of x. Then


Description: 88005.png 


The function that can be easily integrated should be chosen as v and the other function which is easily differentiable should be chosen as u.

Integration by the Method of Partial Fraction

Type 1:


where A and B are constants to be determined.

Type 2:


where AB and C are constants to be determined.

Type 3:


where A and B and C are constants to be determined.

Definite Integration

Let Description: 87988.png where f (x) is the integral of F(x).


As x changes from a to b, the value of the integral changes from f (a) to f (b). This can be shown as


Description: 87984.png 

Properties of Definite Integrals

  • Description: 87980.png
  • Description: 87976.png 
  • Description: 87972.png 
  • Description: 87967.png 
  • Description: 87963.png 
  • Description: 87959.png 
  • Description: 87955.png is an even function
  • Description: 87951.png if f (x) is an odd function

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