An Overview of Differentiation
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:
If f(x) = tan(x) then f’(x) = sec2x.
If f(x) = cos(x) then f’(x) = −sin x.
If f(x) = sin(x) then f’(x) = cos x.
If f(x) = In(x) then f’(x) = 1/x.
If f(x) = ex then f’(x) = ex.
If f(x) = xn then f’(x) = nxn-1 where n is any fraction or integer.
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:
Product Rule
Sum and Difference Rule
Chain Rule
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
FAQs on Concept of Differentiation
1. What is differentiation? Give some real-life differentiation examples.
Differentiation meaning in Maths is defined as a derivative of a function in terms of an independent variable. In Mathematics, it is used to measure the function per unit change in the independent variable. The opposite of differentiation is known as anti-differentiation.
We can find the rate of change of one quantity with respect to another with the help of differentiation. It is useful in many ways.
Some of the real-life differentiation examples are as follows:
Acceleration.
It helps us to derive tangent and normal to a curve.
It helps to calculate the highest or lowest points of a curve in a graph.
2. What are the differentiation formulae in class 11 physics?
The differentiation formulas in Class 11 Physics are as follows. We have to consider f(x) as a function and f’(x) as the derivative of the function:
If f(x) = tan(x) then f’(x) = sec2x.
If f(x) = cos(x) then f’(x) = −sin x.
If f(x) = sin(x) then f’(x) = cos x.
If f(x) = In(x) then f’(x) = 1/x.
If f(x) = ex then f’(x) = ex.
If f(x) = xn then f’(x) = nxn−1 where n is any fraction or integer.
If f(x) = k then f’(x) = 0 and here k is a constant.
3. What is the derivative of cos(x)sin(x) ?
The product rule says:
the derivative of fg = f g’ + f’ g
In our case:
f = cos
g = sin
We know (from the table above):
d/dx of cos(x) = −sin(x)
d/dx of sin(x) = cos(x)
So, the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)
= cos2(x) − sin2(x).
4. Can you explain partial differentiation?
In ordinary differentiation, we find the derivative with respect to one variable only since the function contains one variable. Partial differentiation is used for mathematical functions with more than one variable. Partially differentiated functions are used to find maxima and minima in optimization problems. Partial differentiation is more general than ordinary differentiation. Another name for this is partial derivative. Partial derivatives are derivatives in which we hold some independent variables constant and find the derivatives regarding another independent variable. For example, suppose we have an equation of a curve with two independent variables, X and Y. To determine the slope of the curve in the direction of X, we will take the partial derivative.
5. Explain the concept of higher order differentiation.
Differentiation at the higher order: The successive differentiation of higher order derivatives is demonstrated here. f'" or f'" indicates the further differentiation of the first derivative d2y/dx2. F'" represents the third derivative, and F'" represents the fourth derivative d3y/dx3. A power series is calculated with f n(x), which is the nth derivative of f(x). For example, the displacement rate can be determined by the velocity. In displacement, the second derivative is acceleration, and the third derivative is the jerk.
