[Maths Class Notes] on Concept of Differentiation Pdf for Exam

In Maths, Differentiation in Class 11 is one of the most important topics both academically and in terms of marks weightage. The concept of differentiation refers to the method of finding the derivative of a function. It is the process of determining the rate of change in function on the basis of its variables. The opposite of differentiation is known as anti-differentiation. Suppose, we have two variables x and y. Then, the rate of change of x with respect to y is denoted as dy/dx. The general expression of the derivative of a function is f’(x)= dy/dx where y= f(x) is any function.

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What is Differentiation in Mathematics?

Differentiation in Mathematics is defined as a derivative of a function in terms of an independent variable.

Let f(x) be a function of x and y be another variable.  

Here, the rate of change of y per unit change in x is denoted by dy/dx.

If the function f(x) goes through an infinitesimal change of h near to any point x, the function is defined as,

Lim f(x + h) − f(x)/ h

h tends to 0.

What is Differentiation in Physics?

Differentiation in physics is the same as differentiation in Mathematics. The concepts from differentiation in Maths are used in physics too.

Some Important Formulas in Differentiation

Some important differentiation formulas in Class 11 are given below. We have to consider f(x) as a function and f’(x) as the derivative of the function:

1. If f(x) = tan(x) then f’(x) = sec²x.

2. If f(x) = cos(x) then f’(x) = −sin x.

3. If f(x) = sin(x) then f’(x) = cos x.

4. If f(x) = In(x) then f’(x) = 1/x.

5. If f(x) = e^x then f’(x) = e^x.

6. If f(x) = xⁿ then f’(x) = nx^n-1 where n is any fraction or integer.

7. If f(x) = k then f’(x) = 0 and here k is a constant.

Rules of Differentiation

The main differentiation rules that need to be followed are given below:

1. Product Rule

2. Sum and Difference Rule

3. Chain Rule

4. Quotient Rule

Product Rule – According to the product rule, if the function f(x) is the product of two functions suppose a(x) and b(x), then the derivative of that function is:

If f(x) = a(x) * b(x) then,

f’(x) = a’(x) * b(x) + a(x) * b’(x)

Sum and Difference Rule – According to the sum and difference rule, if the function f(x) is the sum or difference of two functions suppose a(x) and b(x), then the derivative of the function is as follows:

If f(x) = a(x) + b(x) then,

f’(x) = a’(x)+ b’(x)

If f(x) = a(x) − b(x) then,

f’(x) = a’(x) − b’(x)

Chain Rule – If a function y= f(x) = g(u) and if u = h(x), then according to the chain rule for differentiation,

dy/dx = dy/du * du/dx

This rule is very important in the method of substitution during differentiation of composite functions.

Quotient Rule – If the function f(x) is the quotient of two functions i.e. a(x)/b(x), then according to quotient rule, the derivative of the function is as follows:

If f(x) = a(x)/b(x) 

then, f’(x) = a’(x) * b(x) − a(x) * b’(x) / (b(x))²

As a result, for individual functions composed of combinations of these classes, the theory provides the following basic rules for differentiating either the sum, product, or quotient of two functions f(x) and g(x), whose derivative is known (where a and b are constant). D(af + bg) = aDf + bDg (sum); D(fg) = fDg + gDf (product); and D(f/g) = (gDf − fDg)/g² (quotients).

Other basic rules can be applied to composite functions, including the chain rule. By taking the value of g(x) and f(x) as inputs to the composite function f(g(x)), where f(x) = sin x²  , g(f(x)) = (sin x²) , all calculated for a given value of x; for instance, if f(x) = sin x and g(x) = x², then g(f(x)) = (sin x²). The chain rule provides that the derivative of a composite function is equal to the product of the derivatives of the component functions. So, D(f(g(x)) = Df(g(x)) ∙ Dg(x). It is necessary to first find the derivative of Df(x), and x, as necessary, is then replaced by the function g(x). This is the first factor on the right. Using the rule shown above, we get D(sin x² ) = D sin(x²)  ∙ D(x²) = (cos X²) ∙ 2x.

The chain rule takes on the more memorable “symbolic cancellation” form in the notation of German mathematician Gottfried Wilhelm Leibniz, which uses d/dx in place of D to permit differentiation according to variables, such as:

d(f(g(x)))/dx = df/dg ∙ dg/dx.

Solved Example

1. Differentiate f(x) = 9x³ − 6x + 5 with respect to x.

Solution:  Here, f(x) = 9x³ − 6x + 5.

Differentiating both sides w.r.t. x, we get,

f’(x) =(3)(9)x² − 6

f’(x) = 27x² − 6