📄 Differential Calculus(1)
Differential Calculus(1).
This lesson covers core applied mathematics concepts and their agricultural applications for BSc Agriculture learners.
MATHS :: Lecture 05 :: Differential Calculus(1)
DIFFERENTIATION
In all practical situations we come across a number of variables. The variable is one which takes different values, whereas a constant takes a fixed value.
Let x be the independent variable. That means x can take any value. Let y be a variable depending on the value of x. Then y is called the dependent variable. Then y is said to be a function of x and it is denoted by y = f(x)
For example if x denotes the time and y denotes the plant growth, then we know that the plant growth depends upon time. In that case, the function y=f(x) represents the growth function. The rate of change of y with respect to x is denoted by 
and called as the derivative of function y with respect to x.
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This lesson covers core applied mathematics concepts and their agricultural applications for BSc Agriculture learners.
MATHS :: Lecture 05 :: Differential Calculus(1)
DIFFERENTIATION
In all practical situations we come across a number of variables. The variable is one which takes different values, whereas a constant takes a fixed value.
Let x be the independent variable. That means x can take any value. Let y be a variable depending on the value of x. Then y is called the dependent variable. Then y is said to be a function of x and it is denoted by y = f(x)
For example if x denotes the time and y denotes the plant growth, then we know that the plant growth depends upon time. In that case, the function y=f(x) represents the growth function. The rate of change of y with respect to x is denoted by and called as the derivative of function y with respect to x.
| S.No. | Form of Functions | y=f(x) |  |
|---|
- | Power Formula | xn |
- | Constant | C | 0
- | Constant with variable | Cy |
- | Exponential | ex | ex
- | Constant power x | ax | ax log a
- | Logirthamic | logx |
- | Differentiation of a sum| y = u + v
where u and v are functions of x. |
- | Differentiation of a difference | y = u – v
where u and v are functions of x. |
- | Product rule of differentiation | y = uv,
where u and v are functions of x. |
- | Quotient rule of differentiation | y =
,
where u and v are functions of x.
| 
where 
, 
Example
- Differentiate each of the following function
Solution
- Differentiate following function
Solution
Here is the derivative.
- Differentiate following function
Solution
diff. w.r.to x
MPSetEqnAttrs('eq0008','',3,[[90,34,13,-1,-1],[118,46,17,-1,-1],[148,57,21,-1,-1],[],[],[],[372,141,53,-3,-3]]);
ExampleBegin(); 4.__ Differentiate the following functions.
a) 
Solution
MPSetEqnAttrs('eq0009','',3,[[84,18,5,-1,-1],[111,24,7,-1,-1],[139,31,8,-1,-1],[],[],[],[348,77,21,-3,-3]]) MPEquation() 
MPSetEqnAttrs('eq0011','',3,[[76,23,5,-1,-1],[101,32,7,-1,-1],[127,40,8,-1,-1],[],[],[],[315,98,22,-3,-3]]) MPEquation() 
diff y w. r. to x MPSetEqnAttrs('eq0012','',3,[[154,27,9,-1,-1],[205,38,13,-1,-1],[257,47,16,-1,-1],[],[],[],[641,114,39,-3,-3]]) MPEquation() 
MPSetEqnAttrs('eq0012','',3,[[154,27,9,-1,-1],[205,38,13,-1,-1],[257,47,16,-1,-1],[],[],[],[641,114,39,-3,-3]]); MPSetEqnAttrs('eq0013','',3,[[132,62,28,-1,-1],[175,83,37,-1,-1],[218,105,47,-1,-1],[],[],[],[547,258,117,-3,-3]]);
ExampleBegin(); 5. Differentiate the following functions. MPSetEqnAttrs('eq0014','',3,[[218,34,14,-1,-1],[289,45,19,-1,-1],[361,55,23,-1,-1],[],[],[],[905,141,58,-3,-3]]) MPEquation() MPSetEqnAttrs('eq0009','',3,[[84,18,5,-1,-1],[111,24,7,-1,-1],[139,31,8,-1,-1],[],[],[],[348,77,21,-3,-3]]);
MPSetEqnAttrs('eq0010','',3,[[130,17,6,-1,-1],[172,22,7,-1,-1],[216,27,8,-1,-1],[],[],[],[539,71,22,-3,-3]]) MPEquation()
diff f(x) w r to x
Derivatives of the six trigonometric functions
Example
1.__ Differentiate each of the following functions.
Solution We’ll just differentiate each term using the formulas from above.
2.__ Differentiate each of the following functions 
Here’s the derivative of this function.
Note that in the simplification step we took advantage of the fact that
to simplify the second term a little.
3.__ Differentiate each of the following functions 
In this part we’ll need to use the quotient rule.
---|---
Summary Cheat Sheet
- Focus on core formulas, definitions, and solved patterns from this lesson.
- Practice stepwise derivations and numerical substitutions carefully.
- Connect each concept to practical agricultural problem-solving contexts.
References
1 source
References
Primary classroom notes and standard BSc Agriculture applied mathematics references.
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