Derivatives Meaning | First and Second order Derivatives, Formulas and Examples (2024)

Derivatives are defined as the varying rate of change of a function with respect to an independent variable. The derivative is primarily used when there is some varying quantity, and the rate of change is not constant. The derivative is used to measure the sensitivity of one variable (dependent variable) with respect to another variable (independent variable). In this article, we are going to discuss what are derivatives, the definition of derivatives Math, limits and derivatives in detail.

Derivatives Meaning | First and Second order Derivatives, Formulas and Examples (1)

Table of Contents:

  • Meaning
  • Derivatives in Maths
  • Formulas
  • Types
  • Derivatives of Trigonometric Functions
  • Derivative of tanx
  • Derivative of 1 x
  • Properties
  • Examples
  • FAQs

Derivatives Meaning

Derivatives in Maths refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount.

Derivative Example:

Let a car takes ‘t’ seconds to move from a point ‘a’ to ’b’.

Derivatives Meaning | First and Second order Derivatives, Formulas and Examples (2)

But how long will it take to move from point ‘a’ to ‘c’?

Or

How much distance will it cover in ‘t-1’ seconds?

This can be known from the velocity that is as follows:

Velocity (v) = d(x)/d(t)

Where ‘x’ is the distance travelled and ‘t’ is the time taken to cover that distance.

This will give you the distance covered per unit time so that we can analyze any distance covered in any interval of time.

Derivatives in Math – Calculus

The process of finding the derivative is called differentiation. The inverse process is called anti-differentiation. Let’s find the derivative of a function y = f(x). It is the measure of the rate at which the value of y changes with respect to the change of the variable x. It is known as the derivative of the function “f”, with respect to the variable x.

If an infinitesimal change in x is denoted as dx, then the derivative of y with respect to x is written as dy/dx.

Here the derivative of y with respect to x is read as “dy by dx” or “dy over dx”

Example:

Let ‘y’ be a dependent variable and ‘x’ be an independent variable.

Consider a change in the value of x, that is dx.

This change in x will bring a change in y, let that be dy.

Now to find out the change in y with a unit change in x as follows:

Let f(x) be a function whose value varies as the value of x varies

Steps to find the Derivative:

  1. Change x by the smallest possible value and let that be ‘h’ and so the function becomes f(x+h).
  2. Get the change in value of function that is : f(x + h) – f(x)
  3. The rate of change in function f(x) on changing from ‘x’ to ‘x+h’ will be

\(\begin{array}{l}\frac{dy}{dx} = lim_{h\rightarrow 0}\frac{f(x+h) – f(x)}{h}\end{array} \)

Now d(x) is ignorable because it is considered to be too small.

Derivatives Formulas

The formulas of derivatives for some of the functions such as linear, exponential and logarithmic functions are listed below:

  • d/dx (k) = 0, where k is any constant
  • d/dx(x) = 1
  • d/dx(xn) = nxn-1
  • d/dx (kx) = k, where k is any constant
  • d/dx (√x) = 1/2√x
  • d/dx (1/x) = -1/x2
  • d/dx (log x) = 1/x, x > 0
  • d/dx (ex) = ex
  • d/dx (ax) = axlog a

Derivatives Types

Derivatives can be classified into different types based on their order such as first and second order derivatives. These can be defined as given below.

First-Order Derivative

The first order derivatives tell about the direction of the function whether the function is increasing or decreasing. The first derivative math or first-order derivative can be interpreted as an instantaneous rate of change. It can also be predicted from the slope of the tangent line.

Second-Order Derivative

The second-order derivatives are used to get an idea of the shape of the graph for the given function. The functions can be classified in terms of concavity. The concavity of the given graph function is classified into two types namely:

  • Concave Up
  • Concave Down

Calculus-Derivative Example

Let f(x) be a function where f(x) = x2

The derivative of x2 is 2x, that means with every unit change in x, the value of the function becomes twice (2x).

Limits and Derivatives

When dx is made so small that is becoming almost nothing. With Limits, we mean to say that x approaches zero but does not become zero.

Mathematically: for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, (here c ∈ R) we have |f(x) − L| < ε

Key Concepts

  • To differentiate a power of x that is in the denominator, first express it as a power with a negative exponent. Eg.

    \(\begin{array}{l}\frac{1}{x^{2}}= x^{-2}\end{array} \)

  • Derivative rules simplify the process of differentiating polynomial functions.
  • To differentiate a radical, first, express it as a power with a rational exponent

Apply Derivative Rules to Solve an Instantaneous Rate of Change Problem

A skydiver jumps out of a plane from a height of 2200 m. The skydiver’s height above the ground, in meters, after t seconds is represented by the function h(t) = 2200 – 4.9t2(assuming air resistance is not a factor). How fast is the skydiver falling after 4 s?

Solution

The instantaneous rate of change of the height of the skydiver at any point in time is represented by the derivative of the height function.

h(t) = 2200 – 4.9t2

h'(t) = 0 – 4.9 (2t) = -9.8 t

Substitute t = 4into the derivative function to find the instantaneous rate of change at 4 s.

h'(t) = – 9.8 (4) = -39.2

After 4 s, the skydiver is falling at a rate of 39.2 m/s.

Derivatives of Trigonometric Functions

We can also find the derivative of trigonometric functions that means for sin, cos, tan and so on. The formulas are given below:

  • d/dx (sin x) = cos x
  • d/dx (cos x) = -sin x
  • d/dx (tan x) = sec2x
  • d/dx (cosec x) = -cosec x cot x
  • d/dx (sec x) = sec x tan x
  • d/dx (cot x) = -cosec2x

Derivative of tan x

The derivative of tan x can be derived using the quotient rule as shown below:

Let f(x) = tan x

We know that tan x = sin x/ cos x

Let us take u = sin x and v = cos x

As we know,

d/dx (u/v) = [v(du/dx) – u(dv/dx)]/ v2

d/dx (sin x/cos x) = [cos x(d/dx)sin x – sin x(d/dx)cos x]/ cos2x

= [cos x . cos x – sin x . (-sin x)]/cos2x

= (cos2x + sin2x)/cos2x

Using the identity sin2A + cos2A = 1,

= 1/cos2x

= sec2x [since 1/cos x = sec x]

d/dx (tan x) = sec2x

Therefore, the derivative of tan x is sec2x.

Derivative of 1/x

The derivative of 1/x can be derived as given below:

d/dx (1/x) = d/dx (x-1)

We know that d/dx (xn) = nxn – 1

Here, n = -1

d/dx(1/x) = d/dx (x-1) = (-1)x(-1 – 1)

= -x-2

= -1/x2

Hence, the derivative of 1/x is -1/x2.

Properties of Derivatives

Derivatives can be divided into smaller parts so that the given expressions can be easily evaluated. In the process of splitting the expressions or functions, the terms are separated based on the operator such as plus (+), minus (-) or division (/). This can be better understood using the examples given below.

Derivatives Examples

Example 1: Find the derivative of the function f(x) = 5x2– 2x + 6.

Solution:

Given,

f(x) = 5x2– 2x + 6

Now taking the derivative of f(x),

d/dx f(x) = d/dx (5x2– 2x + 6)

Let us split the terms of the function as:

d/dx f(x) = d/dx (5x2) – d/dx (2x) + d/dx (6)

Using the formulas:

d/dx (kx) = k and d/dx (xn) = nxn – 1

⇒ d/dx f(x) = 5(2x) – 2(1) + 0 = 10x – 2

Example 2: Find the derivative of 2 tan x + 1

Solution:

Let the given function be f(x) = 2 tan x + 1

Now, taking the derivative,

d/dx f(x) = d/dx (2 tan x + 1)

= d/dx (2 tan x) + d/dx (1)

= 2 (sec2x) + 0

= 2 sec2x

Frequently Asked Questions – FAQs

Q1

What is derivative in Maths?

In mathematics, derivative is defined as the method that shows the simultaneous rate of change. That means it is used to represent the amount by which the given function is changing at a certain point.

Q2

What are derivatives in finance?

In finance, a derivative is a contract between two or more parties whose value is based on an agreed-upon underlying financial asset or set of assets such as security and index, respectively.

Q3

What is the derivative formula?

The derivative of a function y = f(x) can be expressed as dy/dx = d/dx f(x), where dy/dx is called derivative of y with respect to x and d/dx f(x) is the derivative of f(x) with respect to x.

Q4

What is the derivative symbol?

The symbol used to denote the derivative of a function f(x) is d/dx f(x) or f’(x). If the function is represented using y, then its derivatives of first order and second order are respectively denoted as y’ and y’’.

Q5

Why is it called derivative?

The term derivative is assumed to be derived from the fact that it is another, i.e. the different function f′(x) which is designated by the original function f(x). Thus, f'(x) has been derived from the other function, say f(x).

For more Maths-related concepts, download BYJU’S – The Learning App and also watch interactive videos to learn with ease.
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I'm an expert in mathematical concepts, particularly derivatives, with a deep understanding of the principles and applications. My expertise extends to the mathematical intricacies involved in calculating derivatives, their significance in various fields, and the fundamental concepts associated with them.

Now, let's delve into the concepts mentioned in the article about derivatives:

Meaning: Derivatives in Maths refer to the instantaneous rate of change of a quantity with respect to another. They help investigate the moment-by-moment nature of a quantity.

Derivatives in Math – Calculus: The process of finding a derivative is called differentiation. The derivative of a function (y = f(x)) is the measure of the rate at which the value of (y) changes concerning the change in the variable (x). It is denoted as (\frac{dy}{dx}).

Derivatives Formulas: Various formulas for derivatives were provided, including those for linear, exponential, and logarithmic functions. Notable ones include:

  • (\frac{d}{dx}(k) = 0) (where (k) is any constant)
  • (\frac{d}{dx}(x) = 1)
  • (\frac{d}{dx}(x^n) = nx^{n-1})

Derivatives Types: Derivatives can be classified into first and second order based on their order. First-order derivatives indicate the direction of the function (whether it's increasing or decreasing), while second-order derivatives provide insights into the shape of the graph.

Limits and Derivatives: Limits are discussed in the context of making (dx) almost nothing. The mathematical representation is given, emphasizing the approach of (x) towards zero.

Key Concepts: Tips on differentiating a power of (x) in the denominator, derivative rules simplifying the process, and applying them to solve instantaneous rate of change problems were explained.

Derivatives of Trigonometric Functions: Formulas for finding derivatives of trigonometric functions such as (\sin x), (\cos x), and (\tan x) were provided.

Derivative of tan x: The article walks through the derivation of the derivative of (\tan x) using the quotient rule, ultimately resulting in (\sec^2 x).

Derivative of 1/x: The derivative of (1/x) was derived as (-1/x^2).

Properties of Derivatives: Derivatives can be divided into smaller parts to simplify the evaluation of expressions. The process involves separating terms based on operators like plus (+), minus (-), or division (/).

Derivatives Examples: Examples were given to find the derivative of specific functions using the formulas and rules mentioned earlier.

Frequently Asked Questions (FAQs): Common questions related to derivatives, such as their definition in maths and their application in finance, were addressed.

This summary provides an overview of the key concepts discussed in the article about derivatives. If you have any specific questions or need further clarification on any topic, feel free to ask.

Derivatives Meaning | First and Second order Derivatives, Formulas and Examples (2024)
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