Introduction To Derivatives in Mathematics
In the subject of mathematics, the rate of change of an expression or law that defines the relationship between two variables and provides fundamental solutions to various calculus-related problems and differential equations are known as derivatives. In other words, derivatives are the techniques that help individuals obtain the rate of change in variables and apply them to differential equations to yield an actual result.
Definition of Derivative
The derivative of a function of a real variable in mathematics describes the sensitivity of the function value to a change in its argument. Calculus relies heavily on derivatives. $f^{\prime}$ is the derivative of f(x) and is defined by:
$f^{\prime}=\lim_{h \to 0} \dfrac{f(x+h)-f(x)}h$
Derivatives Used in Mathematics
Gottfried Leibniz and Isaac Newton, during the mid-seventeenth century, invented calculus. Later, derivatives were discovered and utilized for advancement and efficiency in yielding accurate results of variable functions related to the subject of calculus.
The three basic derivatives used in mathematics are mentioned below:
1. For use in algebraic expressions: D(xn) = nxn-1 (where n is a real number)
2. For use in trigonometric functions: D(sin x) = cos x and D(cos x) =(-sin x)
3. For use in exponential functions: D(ex) = ex
In geometrical terms, one can define the derivative of any given function as the inclination of a line to its horizontal plane in a graph of the function or the tendency of inclination of its tangent at a said point.
Use of Derivatives
In the subject matter of mathematics, derivatives play a significant role in the various operations associated with the subject of Calculus. Hence, to learn Calculus thoroughly, an individual should know how to use or apply derivatives in Calculus functions. The following are some methods of using derivatives.
Derivatives can be used to determine the rate of change of quantities.
We can also use them to determine the equations of tangent and normal to a given curve at a point.
The ways to find the turning points on the graph of a function which in return will assist us in determining points at which the largest or smallest value (locally) of any given function occurs.
Derivatives find their uses in Newton’s method. It helps to find the root of a function and determine the concavity of the function, which is whether the function is increasing or decreasing in nature.
An individual can use derivatives to determine intervals on which a given function increases or decreases or find the maxima or minima of the function.
Lastly, the derivative in mathematics can be used to find the approximate value of specific quantities.
Application of Derivatives Formulas
Derivatives have different functionality, such as trigonometric, implicit, logarithmic, etc., as mentioned above. There is an extensive range of subjects in which derivatives can be applied. Derivative formulas find critical usage not only in the field of mathematics but in topics such as science, engineering, computer science, and physics also include the dynamic application of Derivatives in various problems.
In mathematics primarily, derivative formulas are used in the following ways as listed below:
Rate of change of Quantity
Tangent and Normal to a Curve
Newton's Laws
Increasing and Decreasing Functions
Minimum and Maximum values
Linear Approximation
Application of Derivatives in Real Life
Derivatives are used in real-life applications as well for a vivid understanding of various problems and issues. The use of derivatives moreover builds knowledge of the problem by an individual. The application of derivatives in real life are:
Calculating profit and loss in a business scenario in life.
In checking or confirming the variations in temperature conditions.
Determining the distance or speed, or area covered. (In kilometres, miles, etc.).
Derivatives are used in certain equations in Physics.
Derivative formulas are extensively used in Seismology to define the magnitude of earthquakes and for analyzing tectonic activities.
Why Application of Differentiation is Important?
The term differentiation in mathematics is defined as the method that helps us find the rate of change of a specific function or the derivative. The practical technique of differentiation can be used purely based on algebraic manipulations, consisting of the use of three basic derivatives, a piece of knowledge about the manipulation of algebraic functions, and the four rules of operation.
By combining the three types of derivatives as mentioned before, one can build a theory or solution that provides the basic rules for differentiation between the sum, product, or the quotient of any two given functions {such as f(x) and g(x)}.
Solved Examples
1. The tangent to the curve $y=x^{2}-5x+5$ parallel to the line 2y = 4x + 1, also passes through a point. Find the coordinates of the point.
Solution: The curve given is $y=x^{2}-5x+5$.
The slope of the tangent is given by $\dfrac{\text{d}y}{\text{d}x}=2x-5$
In the question, it is given that tangent is parallel to the line,
2y=4x+1
Since the equation of line is 2y=4x+1, the slope of line is $\dfrac{\text{d}y}{\text{d}x}=2$
$\Longrightarrow 2x-5=2$
$\Longrightarrow 2x=7$
$\Longrightarrow x=\dfrac{7}{2}$
Putting the value of x in the equation of curve,
$\Longrightarrow y=\dfrac{49}{4}-5\times\dfrac{7}{2}+5$
$\Longrightarrow y=\dfrac{-1}{4}$
Let us find the equation of tangent with the values we just obtained,
$\Longrightarrow y+\dfrac{1}{4}=2(x-\dfrac{7}{2})$
$\Longrightarrow 2x-\dfrac{29}{4}=y$
Ans: The point which satisfies this equation is $(\dfrac{1}{8},-7)$
2. Let $f(x)=-x^{2}+4x-3$. What will be the maximum value of f(x) between the interval (−1, 1).
Solution: To begin, with x = 2, f′(x) = -2x + 4 = 0. However, because x = 2 is not in the interval, we do not utilize it. As a result, the only two points to be examined are the endpoints, f(-1) =- 8 and f(1) = 0. As a result, the greatest value of f(x) on [1, 1] is f(1) = 0.
Practice Question MCQs
1. The derivative of ln ex equals one. Is the statement correct or incorrect?
True
False
Answer: A) True
2. What is the derivative of y= 99x at x=10?
0
99
990
10
Answer: B) 99
Conclusion
Derivatives have a wide range of applications, not just in mathematics but also in everyday life. Derivatives, for example, have many essential uses in mathematics, such as determining the rate of change of a quantity, determining the approximation value, determining the equation of Tangent and Normal to a Curve, and determining the Minimum and Maximum Values of algebraic expressions. Derivatives are widely utilised in domains such as science, engineering, physics, and so on.
FAQs on Application of Derivatives
1. What are the main applications of derivatives as per the CBSE Class 12 syllabus?
In the CBSE Class 12 Maths syllabus, the primary applications of derivatives are focused on analysing the behaviour of functions. The key applications include:
- Rate of Change: Determining the rate at which one variable changes with respect to another, such as calculating the speed of a moving object.
- Increasing and Decreasing Functions: Identifying the intervals where a function's value is rising or falling.
- Tangents and Normals: Finding the equations of the tangent line and the normal line to a curve at a specific point.
- Approximations: Using derivatives to find the approximate value of a function near a known point.
- Maxima and Minima: Locating the points where a function reaches its highest (maximum) or lowest (minimum) value, which is crucial for solving optimisation problems.
2. How do derivatives help in calculating the rate of change of quantities?
Derivatives measure the instantaneous rate of change of a quantity. If a quantity y is a function of another quantity x, represented as y = f(x), then its derivative, dy/dx or f'(x), gives the rate at which y changes with respect to x. For example, if s(t) represents the position of an object at time t, its derivative, v(t) = ds/dt, represents the object's instantaneous velocity.
3. How can you use the first derivative to determine if a function is increasing or decreasing?
The first derivative test is used to determine where a function is increasing or decreasing. The sign of the first derivative, f'(x), indicates the slope of the tangent to the function's graph:
- If f'(x) > 0 on an interval, the function is increasing on that interval.
- If f'(x) < 0 on an interval, the function is decreasing on that interval.
- If f'(x) = 0 at a point, that point is a stationary point, which could be a local maximum, minimum, or a point of inflection.
4. What is the role of derivatives in finding the equations for a tangent and a normal to a curve?
The derivative of a function at a point gives the slope of the tangent to the curve at that point. To find the equations:
- Calculate the derivative f'(x) of the function y = f(x).
- Evaluate the derivative at the given point (x₁, y₁) to find the slope of the tangent, m = f'(x₁).
- The equation of the tangent is given by the point-slope form: (y - y₁) = m(x - x₁).
- The normal is perpendicular to the tangent, so its slope is -1/m. The equation of the normal is: (y - y₁) = (-1/m)(x - x₁).
5. Explain how derivatives are used to find the local maximum and minimum values of a function.
Derivatives are used to find local maxima and minima by identifying critical points where the function's slope is zero. The process involves:
- Step 1: Find the first derivative, f'(x), of the function.
- Step 2: Set f'(x) = 0 and solve for x to find the critical points (or stationary points).
- Step 3: Use the second derivative test. Find the second derivative, f''(x). For each critical point 'c':
- If f''(c) < 0, the function has a local maximum at x = c.
- If f''(c) > 0, the function has a local minimum at x = c.
- If f''(c) = 0, the test is inconclusive, and one must revert to the first derivative test.
6. Are the 'derivatives' in finance the same as the derivatives studied in calculus?
No, they are fundamentally different concepts that happen to share the same name. In calculus, a derivative is a mathematical tool that measures the rate of change of a function. In finance, a derivative is a contract or financial instrument whose value is derived from an underlying asset like stocks, bonds, or commodities. While advanced financial modelling can use calculus, the term 'derivative' in a finance context does not refer to the mathematical operation of differentiation.
7. Why is the second derivative test useful for identifying maxima and minima? What does its sign signify?
The second derivative test is useful because it describes the concavity of the function's graph at a stationary point. The sign of the second derivative, f''(x), tells us how the slope (the first derivative) is changing:
- A negative second derivative (f''(x) < 0) means the slope is decreasing. The graph is concave down (like an upside-down 'U'), indicating a local maximum.
- A positive second derivative (f''(x) > 0) means the slope is increasing. The graph is concave up (like a 'U'), indicating a local minimum.
Essentially, the second derivative provides a quick way to determine the shape of the curve at a point where the tangent is horizontal.
8. What is a practical example of using derivatives to solve an optimisation problem in real life?
A classic real-life optimisation problem involves maximising an area. For example, if you have a fixed length of fencing and want to enclose the largest possible rectangular garden, derivatives can find the optimal dimensions. You would create a function for the area in terms of one of its dimensions, find the derivative of this function, set it to zero to find the critical point, and then use the second derivative test to confirm that this point yields the maximum possible area. This principle is used in manufacturing to minimise material costs and in business to maximise profit.
