Definition Formula and Solved Examples 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 in Calculus
1. What is differentiation in calculus?
Differentiation is the process of finding the derivative of a function, which measures how the function changes with respect to a variable. In calculus, it represents the rate of change or slope of the tangent line to a curve at a given point.
- If y = f(x), then its derivative is written as dy/dx or f'(x).
- It tells how fast y changes when x changes.
- It is a fundamental concept in differential calculus.
2. What is the derivative of a function?
The derivative of a function is the instantaneous rate of change of the function with respect to its variable. It is defined using limits as:
- f'(x) = lim(h→0) [f(x + h) − f(x)] / h
3. What is the formula for differentiation of xⁿ?
The differentiation formula for xⁿ is given by the power rule: d/dx (xⁿ) = n·xⁿ⁻¹.
- Here, n can be any real number.
- Example: d/dx (x³) = 3x².
- Example: d/dx (x⁵) = 5x⁴.
4. How do you differentiate a constant?
The derivative of any constant is 0.
- If f(x) = c, where c is a constant, then d/dx (c) = 0.
- This is because a constant does not change as x changes.
- Example: d/dx (7) = 0.
5. What is the difference between differentiation and integration?
Differentiation finds the rate of change, while integration finds the accumulated quantity or area under a curve.
- Differentiation gives the slope or derivative of a function.
- Integration gives the antiderivative or total accumulation.
- They are inverse processes in calculus.
6. What is the geometric meaning of differentiation?
The geometric meaning of differentiation is the slope of the tangent line to a curve at a given point.
- If y = f(x), then f'(x) gives the slope at any point x.
- A positive derivative means the function is increasing.
- A negative derivative means the function is decreasing.
7. How do you differentiate a sum of functions?
To differentiate a sum of functions, differentiate each term separately using the sum rule.
- If f(x) = u(x) + v(x), then f'(x) = u'(x) + v'(x).
- Example: d/dx (x² + 3x) = 2x + 3.
8. What is the product rule in differentiation?
The product rule is used to differentiate the product of two functions and is given by d/dx (uv) = u·dv/dx + v·du/dx.
- If y = u(x)v(x), then y' = u·v' + v·u'.
- Example: d/dx (x·x²) = x·2x + x²·1 = 3x².
9. What is the chain rule in differentiation?
The chain rule is used to differentiate a composite function and is given by d/dx [f(g(x))] = f'(g(x)) · g'(x).
- It applies when one function is inside another.
- Example: If y = (x² + 1)³, then dy/dx = 3(x² + 1)² · 2x = 6x(x² + 1)².
10. What are the applications of differentiation in real life?
Differentiation is used to calculate rates of change, optimize values, and analyze motion in real-life problems.
- In physics, it finds velocity and acceleration.
- In economics, it helps determine marginal cost and marginal revenue.
- In engineering, it is used for optimization and curve modeling.
