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 

8. 
Differentiation of a difference 
y = u – v 

9. 
Product rule of differentiation 
y = uv, 

10. 
Quotient rule of differentiation 
y = , 

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.
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