Key Differentiation Rules and Problem-Solving Tips
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 Differentiation in Maths: Concepts and Applications
1. What is the core concept of differentiation in Maths?
In mathematics, differentiation is the process of finding the instantaneous rate of change of a function with respect to one of its variables. Imagine you are in a car; your average speed is total distance divided by total time, but your speedometer shows your speed at a specific instant. Differentiation gives you that instantaneous value. Geometrically, it calculates the slope of the tangent line to a function's graph at any given point.
2. What are some real-world examples of the application of differentiation?
Differentiation is a fundamental concept used across various fields. Here are a few examples:
- Physics: To calculate instantaneous velocity and acceleration from a position function.
- Economics: To find the marginal cost and marginal revenue, which helps in maximising profit.
- Engineering: To determine the rate of change of quantities, such as the rate of cooling of an object or the rate of flow of a liquid.
- Geometry: To find the equations of tangents and normals to a curve at a specific point.
3. What are the basic differentiation formulas every Class 11 student should know?
For students beginning with the concept of differentiation as per the CBSE/NCERT syllabus, mastering a few basic formulas is essential:
- Power Rule: The derivative of xn is nxn-1.
- Constant Rule: The derivative of any constant (e.g., 5) is 0.
- Exponential Functions: The derivative of ex is ex, and the derivative of ax is axlog(a).
- Logarithmic Function: The derivative of log(x) is 1/x.
- Trigonometric Functions: The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x).
4. How is the derivative of a function related to the slope of its tangent?
The derivative of a function at a specific point provides the exact value of the slope of the tangent line to the function's graph at that same point. If you calculate the derivative f'(x) at x = a, the result, f'(a), is the gradient of the straight line that just touches the curve at x = a. This geometric interpretation is a cornerstone of understanding differentiation.
5. What is the difference between a derivative and differentiation?
While often used interchangeably, these terms have distinct meanings. Differentiation refers to the entire process or operation of finding the derivative. It's the action you perform. The derivative, on the other hand, is the result or output of this process. It is a new function that describes the rate of change of the original function.
6. How does the Product Rule work in differentiation? Can you provide an example?
The Product Rule is used to find the derivative of a function that is the product of two other functions. The rule is: if h(x) = f(x)g(x), then its derivative h'(x) = f(x)g'(x) + g(x)f'(x). In simple terms, it's the first function times the derivative of the second, plus the second function times the derivative of the first. For example, to differentiate x²sin(x), you would get x²(cos(x)) + sin(x)(2x).
7. Why is the derivative of a constant function always zero?
The derivative measures the rate of change or the slope of a function. A constant function, such as y = 5, represents a perfectly horizontal line on a graph. By definition, a horizontal line has no steepness or slope at any point. Since its slope is consistently zero everywhere, its rate of change is also zero. Therefore, its derivative must be 0.
8. What is a 'higher-order derivative' in the context of Class 12 Maths?
A higher-order derivative is the result of differentiating a function more than once. The first derivative, f'(x), gives the slope. If you differentiate f'(x) again, you get the second derivative, denoted as f''(x) or d²y/dx². In the Class 12 syllabus, the second derivative is crucial for determining the concavity of a curve (whether it bends upwards or downwards) and for finding points of inflection.
9. How does differentiation help in finding the maximum or minimum value of a function?
Differentiation is the primary tool for solving optimization problems. At a local maximum or minimum point of a smooth curve, the tangent line is horizontal. A horizontal line has a slope of zero. Since the first derivative, f'(x), gives the slope of the tangent, we can find these potential maximum or minimum points by setting the first derivative equal to zero (f'(x) = 0) and solving for x. This is the first step in the 'First Derivative Test'.
10. What is the physical meaning of the first and second derivatives in motion?
In physics, derivatives provide a clear description of motion. If a function s(t) describes an object's position at time t:
- The first derivative, s'(t), represents the object's instantaneous velocity (how fast it is moving and in what direction).
- The second derivative, s''(t), represents the object's instantaneous acceleration (the rate at which its velocity is changing).
