![Page 1: 9. Basic Concepts of Differential and Integral Calculus](https://reader036.vdocuments.us/reader036/viewer/2022082507/563dbb45550346aa9aabc05c/html5/thumbnails/1.jpg)
9. Basic Concepts of Differential and Integral Calculus
DifferentiationDifferentiation is the process of finding rate of change of a dependent variable with respect to independent variable.
Differentiation Formulae
( )y f x
Dependent Variable
Independent Variable
Also remember the following formulae:
![Page 2: 9. Basic Concepts of Differential and Integral Calculus](https://reader036.vdocuments.us/reader036/viewer/2022082507/563dbb45550346aa9aabc05c/html5/thumbnails/2.jpg)
Differentiation Techniques
1.
Exp:
2.
Exp:
3.
Exp:
4. Product Rule:
In general,
Exp:
5. Quotient Rule:
Exp:
6. Derivative of a function of function (Chain Rule):
If
Exp: If , then find .
Solution: Let .
7. Derivative of Implicit Functions:
![Page 3: 9. Basic Concepts of Differential and Integral Calculus](https://reader036.vdocuments.us/reader036/viewer/2022082507/563dbb45550346aa9aabc05c/html5/thumbnails/3.jpg)
A function in the form of e.g. where y cannot be directly defined as a function of x is called an implicit function of x.
Exp: If , then find .
Solution:
8. Derivative of Parametric Equation:When both the variables are expressed in terms of a parameter (a third variable), then the involved equations are called as parametric equations.
Exp: If , then find .
Solution:
9. Logarithmic Differentiation:This procedure of finding out derivative by taking logarithm is used in the following two situations:
.
When the function is the product of number of functions.
Exp: If , then find .
Solution:
![Page 4: 9. Basic Concepts of Differential and Integral Calculus](https://reader036.vdocuments.us/reader036/viewer/2022082507/563dbb45550346aa9aabc05c/html5/thumbnails/4.jpg)
X
Y
O
P(x, y)
Exp: If , then find .
Solution:
Higher Order Differentiation:
Exp: If , then find .
Solution:
Geometric Meaning of the Derivative
The derivative of at a point x represents the slope or gradient of the tangent to the curve at the point x.
![Page 5: 9. Basic Concepts of Differential and Integral Calculus](https://reader036.vdocuments.us/reader036/viewer/2022082507/563dbb45550346aa9aabc05c/html5/thumbnails/5.jpg)
Integration
Integration is the reverse process of differentiation.
Integration Formulae
( )f x f x
Differentiation
Integration
Also remember the following formulae:
![Page 6: 9. Basic Concepts of Differential and Integral Calculus](https://reader036.vdocuments.us/reader036/viewer/2022082507/563dbb45550346aa9aabc05c/html5/thumbnails/6.jpg)
Integration by parts
Priority of functions to be considered as a first function is as follows:Logarithmic Functions e.g. Algebric Functions e.g. Exponential Functions e.g.
Definite Integration
Important Property of definite Integral