Applications of Derivatives
Derivatives have got several applications such as finding the concavity of a function, finding the slope of tangent and normal, and finding the maxima and minima of a function. Let’s learn them briefly:
Critical Point
Critical Point of a function is the point where the derivative of a function is either zero or not defined. Hence, if P is a critical point of the function then
dy/dx at P = 0 or dy/dx at P = Not Defined
Concavity of a Function
Concavity of a function simply means the opening of the curve of a function is upwards or downwards.
- If the opening of the curve is upwards then it is called Concave Up and if downwards it is called Concave Down.
- The condition for concave up is f”(x) > 0 and the condition for concave down is f”(x) < 0.
- The point at which the concavity of a function changes is called its Inflection Point.
Increasing and Decreasing Function
A function is said to be increasing if for point x < y, f(x) ≤ f(y), and if for the point x > y , the value f(x) ≥ f(y) then the function is said to be decreasing.
- To find if the given function is increasing or not we can test it by using derivative.
- If f'(x) ≥ 0 in the interval then the nature of the function is increasing in the interval.
- If f'(x) ≤ 0 in the interval then the nature of the function is decreasing in the interval.
Slope of Tangent and Normal
Tangent is a line that touches the curve at one point. The slope of the tangent at point P is given as dy/dx at x = P.
A normal is a line that intersects the curve and is perpendicular to the tangent at the point of contact. The slope of normal is given by -1/slope of tangent.
Maxima and Minima
Derivative is used to find the maximum and minimum value of a function.
- There are two types of maximum and minimum points named as absolute maxima and absolute minima and local maxima and local minima.
- In the case of absolute we look for maximum or minimum at a particular point let’s say x = a and for local maxima and minima, we look for maximum and minimum at all points near a.
The condition for these are tabulated below:
Maxima and Minima | ||
|---|---|---|
| Absolute Maximum | x = a | f(x) ≤ f(a) where x and a belong to domain of f. |
| Absolute Minimum | x = a | f(x) ≥ f(a) where x and a belong to domain of f. |
| Local Maximum | x = a | f(x) ≤ f(a) for all x near to a. |
| Local Minimum | x = a | f(x) ≥ f(a) for all x near to a. |
The maxima and minima can be found using two types of derivative tests named first derivative test and second derivative test. Let’s go through them briefly:
First Derivative Test
First Derivative Test involves differentiating the function one time.
The condition for local maxima and local minima is tabulated below:
Local Minima | Local Maxima |
|---|---|
If x = a is point of local minima then
| If x = a is the point of local maxima then
|
Second Derivative Test
Second Derivative Test involves second-order derivatives of the function. The condition for maxima and minima using second order derivative at point x = a is tabulated below:
Second Derivative Test | |
|---|---|
| f”(a) < 0 | x = a is point of local maxima |
| f”(a) > 0 | x = a is point of local minima |
| f”(a) = 0 | x = a may be a point of local maxima or local minima or none. |
Derivatives | First and Second Order Derivatives, Formulas and Examples
Derivatives: In mathematics, a Derivative represents the rate at which a function changes as its input changes. It measures how a function’s output value moves as its input value nudges a little bit. This concept is a fundamental piece of calculus. It is used extensively across science, engineering, economics, and more to analyze changes.
Derivative
Table of Content
- What are Derivatives?
- Derivatives Meaning
- Derivative by First Principle
- Types of Derivatives
- First Order Derivative
- Second Order Derivative
- nth Order Derivative
- Derivatives Formula
- Rules of Derivatives
- Derivative of Composite Function
- Chain Rule of Derivatives
- Derivative of Implicit Function
- Parametric Derivatives
- Higher Order Derivatives
- Partial Derivative
- Logarithmic Differentiation
- Applications of Derivatives
- Derivatives Examples
- Sample Problems on Derivatives
- Practice Problems on Derivatives
A derivative is a calculus tool that measures the sensitivity of a function’s output to its input. It is also known as the instantaneous rate of change of a function at a given position.
The derivative of a function with just one variable is the slope of the line that is tangent to the function’s graph for a given input value. In terms of geometry, the derivative of a function can be defined as the slope of its graph.
In this article we have covered Meaning of Derivatives along with types of derivatives, examples, formulas, applications and many more.
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