💻 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.

S.No.	Form of Functions	y=f(x)

1. | Power Formula | xn |
2. | Constant | C | 0
3. | Constant with variable | Cy |
4. | Exponential | ex | ex
5. | Constant power x | ax | ax log a
6. | Logirthamic | logx |
7. | Differentiation of a sum| y = u + v where u and v are functions of x. |
8. | Differentiation of a difference | y = u – v where u and v are functions of x. |
9. | Product rule of differentiation | y = uv, where u and v are functions of x. |
10. | Quotient rule of differentiation | y = , where u and v are functions of x. | where ,

Example

1. Differentiate each of the following function

Solution

1. Differentiate following function

Solution

Here is the derivative.

1. 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

Primary classroom notes and standard BSc Agriculture applied mathematics references.