Key Formulas Methods and Real Life Applications of Derivatives in Maths
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 Explained with Concepts and Examples
1. What is the application of derivatives in calculus?
The application of derivatives refers to using derivatives to solve real-world problems involving rate of change, slope, optimization, and motion. In calculus, derivatives are applied to:
- Find the slope of a tangent line to a curve.
- Calculate maximum and minimum values (optimization problems).
- Determine velocity and acceleration in motion.
- Analyze increasing/decreasing functions and concavity.
2. How do you find the maximum and minimum values using derivatives?
To find maximum and minimum values, first compute the derivative and set it equal to zero to find critical points. Follow these steps:
- Find the derivative f'(x).
- Solve f'(x) = 0 to get critical points.
- Use the second derivative test:
- If f''(x) > 0, it is a minimum.
- If f''(x) < 0, it is a maximum.
3. What is the formula for the derivative in applications of motion?
In motion, velocity is the derivative of displacement and acceleration is the derivative of velocity. The key formulas are:
- v(t) = ds/dt (velocity)
- a(t) = dv/dt = d²s/dt² (acceleration)
4. How are derivatives used to find the slope of a tangent line?
The slope of a tangent line at a point is equal to the value of the derivative at that point. To find it:
- Compute the derivative f'(x).
- Substitute the given x-value.
5. What are critical points in the application of derivatives?
A critical point is a point where f'(x) = 0 or where the derivative does not exist. These points are important because:
- They help locate local maxima and minima.
- They indicate possible turning points.
6. How do derivatives help in determining increasing and decreasing functions?
A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0. Steps:
- Find the derivative f'(x).
- Determine the sign of f'(x) in different intervals.
7. What is the second derivative test in applications of derivatives?
The second derivative test determines whether a critical point is a maximum or minimum. After finding a critical point:
- If f''(x) > 0, the function has a local minimum.
- If f''(x) < 0, the function has a local maximum.
- If f''(x) = 0, the test is inconclusive.
8. How are derivatives used in real-life applications?
Derivatives are used in real life to measure rates of change and solve optimization problems. Common applications include:
- Physics: calculating velocity and acceleration.
- Economics: maximizing profit and minimizing cost.
- Engineering: optimizing design and efficiency.
- Biology: modeling growth rates.
9. Can you give an example of an optimization problem using derivatives?
An optimization problem using derivatives involves finding the maximum or minimum value of a function. Example: Find two numbers whose sum is 10 and product is maximum.
- Let one number be x, the other 10 − x.
- Product P = x(10 − x) = 10x − x².
- P'(x) = 10 − 2x.
- Set 10 − 2x = 0 → x = 5.
10. What is the difference between first derivative and second derivative applications?
The first derivative measures rate of change and slope, while the second derivative measures concavity and acceleration. Specifically:
- First derivative (f'(x)): slope, increasing/decreasing functions, velocity.
- Second derivative (f''(x)): concavity, maximum/minimum test, acceleration.
