Application Of Derivatives Class 12 Definition Formulas And Solved Examples
This chapter Application of derivatives mainly features a set of topics just like the rate of change of quantities, Increasing and decreasing functions, Tangents and normals, Approximations, Maxima and minima, and lots more. We are going to discuss the important concepts of the chapter application of derivatives.
1. A function f is said to be
Increasing only if on an interval of (a, b) if x1 < x2
Decreasing on (a,b) if x1 < x2
Let the constant (a, b), if f (x) equals c for all x ∈ (a, b), where c is the constant.
2. First Derivative Test.
3. Second Derivative Test.
The main topics which are covered for the NCERT Solutions for Application of Derivatives Class 12 notes, Chapter 6 are:
NCERT Solutions for Application of Derivatives Class 12 Notes Mathematics Chapter 6
Maximum and Minimum Value
Let f be the function which is defined on an interval I. So,
f is claimed to possess a maximum value in I, if there exists some extent c in I such: f(c) > f(x), ∀ x ∈ I. The number f(c) is named the utmost value of f in I and therefore the point c is named to some extent a maximum value of f in I.
f is said to be having a minimum value in I, and if there exists a point c in I so f(c) < f(x), ∀ x ∈ I. The number f(c) is named the minimum value of f in I and therefore the point c is named to some extent the minimum value of f in I.
f is claimed to possess an extreme value in I, if there exists some extent c in I such f(c) is either a maximum value or a minimum value of f in I. The number f(c) is named an extreme value off in I and therefore the point c is named an extreme.
Important Points of Applications of Derivatives
Through the graphs, we will even find the maximum/minimum value of a function to some extent at which it's not even differentiable.
Every monotonic function makes sure that its maximum/minimum value is at the endpoints of the domain of the definition of the function.
Every continuous function on a bounded interval features a maximum and a minimum value.
Let f be a function which is defined on an unbounded interval which is I. Suppose cel is any point. If f has local maxima or local minima at x = c, then either f'(c) = 0 or f isn't differentiable at c.
Critical Point: to some extent c within the domain of a function f at which either f'(c) = 0 or f isn't differentiable, is named a juncture of f.
First Derivative Test: Let f be a function defined on an unbounded interval which is I and f be the continuous of a juncture c in I. So, if f'(x) changes sign from positive to negative as x increases through c, then c may be a point of local maxima.
if f'(x) changes sign from negative to positive as x increases through c, then c may be a point of local minima.
if f'(x) doesn't change sign as x increases through c, then c is neither some extent of local maxima nor some extent of local minima. Such some extent is named some extent of inflection.
Second Derivative Test: Let f(x) be a function that is defined on an interval known as I and c ∈ I. Let f be two times differentiable at c. So,
x = c is a point for the local maxima, if f'(c) equals 0 and f''(c) < 0.
x = c is a point for the local minima, if f'(c) equals 0 and f''(c) > 0.
The test will fail if f'(c) equals 0 and f''(c) equals 0.
FAQs on Application Of Derivatives For Class 12 Explained Clearly
1. What is Application of Derivatives in Class 12 Maths?
The Application of Derivatives in Class 12 Maths refers to using derivatives to solve real-life problems related to rate of change, tangents and normals, increasing–decreasing functions, maxima and minima. It connects differentiation to practical concepts such as optimization and motion. Key areas include:
- Rate of change of quantities
- Increasing and decreasing functions
- Local maxima and minima
- Tangents and normals to curves
2. What is the geometrical meaning of a derivative?
The geometrical meaning of a derivative is that it represents the slope of the tangent to a curve at a given point. If a function is y = f(x), then its derivative dy/dx gives the slope of the tangent at any point. For example, if f(x) = x², then f'(x) = 2x, so at x = 1, slope = 2.
3. How do you find the equation of a tangent to a curve?
The equation of a tangent is found using the formula y − y₁ = m(x − x₁), where m is the derivative at the point. Steps:
- Differentiate y = f(x) to get dy/dx.
- Substitute x = x₁ to find slope m.
- Use point-slope form with point (x₁, y₁).
Example: For y = x² at x = 1, slope = 2. Tangent: y − 1 = 2(x − 1).
4. What is the formula for the equation of normal?
The equation of the normal is given by y − y₁ = (−1/m)(x − x₁), where m is the slope of the tangent. Since the normal is perpendicular to the tangent, its slope is −1/m. Example: If slope of tangent is 2, slope of normal = −1/2.
5. How do you find intervals where a function is increasing or decreasing?
A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0. Steps:
- Find the first derivative f'(x).
- Solve f'(x) = 0 to get critical points.
- Check the sign of f'(x) in each interval.
If the derivative is positive, the function rises; if negative, it falls.
6. What are maxima and minima in Application of Derivatives?
Maxima and minima are the extreme values of a function where it reaches highest or lowest points locally. A point x = a is:
- Local maximum if f'(a) = 0 and f''(a) < 0
- Local minimum if f'(a) = 0 and f''(a) > 0
This is called the second derivative test.
7. How do you solve optimization problems using derivatives?
Optimization problems are solved by finding the maximum or minimum value using the first derivative test. Steps:
- Form an equation for the quantity to optimize.
- Differentiate the function.
- Set derivative equal to zero.
- Use second derivative test to confirm maxima or minima.
This method is widely used in profit maximization and cost minimization problems.
8. What is the first derivative test?
The first derivative test determines maxima or minima by checking sign changes of f'(x) around critical points. If f'(x) changes:
- From positive to negative → Local maximum
- From negative to positive → Local minimum
It analyzes increasing and decreasing behavior near the critical point.
9. What is rate of change in Application of Derivatives?
The rate of change measures how one quantity changes with respect to another and is given by the derivative. For example, if s(t) is distance, then ds/dt represents velocity. If s(t) = t², then velocity = 2t.
10. What are critical points in derivatives?
Critical points are values of x where f'(x) = 0 or f'(x) is not defined. These points are used to find:
- Local maxima
- Local minima
- Points of horizontal tangent
They play a key role in studying increasing–decreasing functions and optimization problems in Class 12 Application of Derivatives.
