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.

Table of Content

• What are Derivatives?
  • Derivatives Meaning
  • Derivative by First Principle
  • First Order Derivative
  • Second Order Derivative
  • nth Order Derivative
  • Chain Rule of Derivatives
  • Higher Order 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.

  What are Derivatives?

  Derivative is defined as the rate of the instantaneous change in a quantity with respect to another quantity.

  Let’s say f is a real-valued function and ‘a’ is a point in its domain of definition. The derivative of f at a is defined as,

  The above statement is subject to the condition that its limits exist. This is also referred to as [Tex]\left.\dfrac\right|_[/Tex]

  Derivatives Meaning

  A derivative represents the rate at which a function is changing at any given point. It is a fundamental concept in calculus and is used to understand how a function behaves as its input changes.

  Derivative by First Principle

  The derivative defined as the limit is called the Derivative by First Principle . Derivative by First Principle is also called Derivative by Delta Method.

  Let’s understand the Derivative by First Principle with the help of the image attached below:

  

  • The diagram given above represents the geometric interpretation of the derivative.
  • Lets we have two points P(a, f(a)) and Q(a + h, f(a + h)) which are close to each other on the graph. We know that according to the definition,
  • From the triangle PQR, it is clear that the ratio whose limit we are taking is precisely equal to tan(QPR) which is the slope of the chord PQ. In the limiting process, as h tends to 0, the point Q tends to P, and we have,
  • We can see that the chord PQ tends to be tangent to the curve f(x). The limit is equal to the slope of the tangent to the curve at a particular point.

  Types of Derivatives

  Below are the different types of Derivatives

  • First Order Derivative
  • Second Order Derivative
  • nth Order Derivative, based on the number of times they are differentiated.

  Let’s learn all the types of derivatives in detail.

  First Order Derivative

  First Order Derivative of a Function is defined as the rate of change of a dependent variable with respect to an independent variable.

  • First Order Derivative gives the slope of the tangent drawn to the curve.
  • It tells about the direction of function and explains if the function is increasing or decreasing in nature.
  • The First Order Derivative can be explained in terms of Limit as follows.

  f'(x) = lim_x→a f(x) – f(a) / x – a

  • The other notations of First Order Derivative are given as dy/dx, D(y), d(f(x))/dx, and y’.

  Second Order Derivative

  Second Order Derivative is the derivative of First Order Derivative of a function. It is also called Derivative of Derivative.

  • It basically means differentiating a function twice successively.
  • The Second Order derivative of a function tells about how the slope of the curve of a function changes.
  • Second Order Derivative gives an idea about the local maxima and local minima of a curve.
  • It is represented as d/dx = d 2 y/dx 2 = f”(x)

  n th Order Derivative

  nth Order Derivative refers to finding successive differentiation of a function ‘n’ number of times. It is represented as d y n/dx n = f n (x).

  Derivatives Formula

  To find the derivative of different functions we need to learn different formulas. However, the basic rule to find the derivative by first principle is valid to all but as it can be too extensive sometimes so we refer to formulas for instant differentiation.

  Some of the most important formulas for derivatives are discussed as follows:

  Power Rule of Derivatives

  Power Rule of Derivatives states that If a function y = f(x) = x n then its derivative

  dy/dx = f'(x) = nx n-1

  where n is an integer

  For example, the derivative of f(x) = x 3 is 3x (3-1) = 3x 2 .

  Derivative of Exponential Function

  The derivative of Exponential Function is listed below:

  • d(e x )/dx = e x
  • d(a x ) = a x ln a

  Where e is the Euler’s number and a is any real positive number.

  Derivative of Logarithmic Function

  • d(ln x)/dx = 1/x
  • d(log _a x)/dx = 1/(x ln a)

  Where ln is the natural logarithm i.e., log with base e [Eular’s Number]

  Derivatives of Trigonometric Functions

  The derivatives of various trigonometric functions are listed below:

  • d(sin x)/dx = cos x
  • d(cos x)/dx = -sin x
  • d(tan x)/dx = sec 2 x
  • d(cot x)/dx = -cosec 2 x
  • d(sec x)/dx = sec x tan x
  • d(cosec x)/dx = -cosec x cot x

  Derivative of Inverse Trigonometric Functions

  If x = sin y then y = sin -1 x is the inverse trigonometric function.

  • d(sin -1 x)/dx = 1/√(1 – x 2 )
  • d(cos -1 x)/dx = -1/√(1 – x 2 )
  • d(tan -1 x)/dx = 1/(1 + x 2 )
  • d(cosec -1 x)/dx = -1/|x|√(x 2 – 1)
  • d(sec -1 x)/dx = 1/|x|√(x 2 – 1)
  • d(cot -1 x)/dx = -1/(1 + x 2 )

  Rules of Derivatives

  There are certain rules to be followed while finding the derivatives of functions. Let’s learn from them

  Power Rule of Derivative

  • If f(x) = x n then d(f(x))/dx = dx n /dx = nx n-1

  Sum and Difference Rule of Derivative

  If two functions are expressed as sum or difference then its derivative is equal to the sum and difference of derivatives of individual function. Let’s say two functions u and v are expressed as u ± v then

  d(u ± v )/dx = du/dx ± dv/dx

  Constant Multiple Rule of Derivative

  If a constant ‘c’ is multiplied to a function f(x) expressed as c.f(x) then the derivative of c.f(x) is given as

  Product Rule of Derivative

  If two functions u and v are given in product form i.e. u.v then its derivative is given as

  Quotient Rule of Derivative

  If two functions are given as quotient form i.e. u/v then its derivative is given as

  d(u/v)/dx = (v.du/dx – u.dv/dx)v 2

  Derivative of Composite Function

  Composite Function is defined as the function of a function. Let’s say we have function f which is a function of another function g(x) then the composite function is written as f(g(x)) or fog(x). Let’s say we have a function y = sin 2 x then we can find the derivative in the following manner:

  Step 1: First assume one function equal to some other variable i.e. u = sin x (let). There for original function becomes y = u 2 where u = sin x

  Step 2: Now derivative of u = sin x with respect to x and y = u 2 with respect to u. Therefore we have du/dx = cos x and dy/du = 2u

  Step 3: Now multiply the two derivatives i.e. (du/dx)(dy/du) = 2u cos x.

  Step 4: Now replace the assumed value u = sin x from step 1.

  Hence, the derivative of sin 2 x is 2sin x cos x.

  Chain Rule of Derivatives

  Chain Rule allows us to differentiate Composite Functions in a single line. In the chain rule, we differentiate functions and write them in product format. For Example if f(x) = sin 2 x then f'(x) = 2sin x cos x. In this example we first differentiated sin 2 x to 2sin x using the Power Rule of Derivative and then differentiated sin x to cos x and wrote both derivatives in product format.

  Derivative of Implicit Function

  When a function is expressed in terms of two variables rather than one single variable then it is called an implicit function. Implicit Differentiation involves the use of a chain rule to differentiate a function. For example, if you have to find the derivative of 3xy2 then its derivative with respect to x is given as d(3xy 2 )/dx = 3y 2 + 3x.2y.dy/dx.

  So we see that we differentiated each variable and wrote it in summation form. However if we have function in the form of f(x,y) = 0 for example 3x + y = 5 then we differentiate it as

  Parametric Derivatives

  If two dependent variables x and y are dependent on a third independent variable let say ‘t’ expresses as x = f(t) and y = g(t), then the function is said to be in Parametric Form.

  • To Find the Parametric Derivative first differentiate the two variables x and y with respect to the independent variable ‘t’ and then take the ratio of their derivatives i.e. first find dx/dt and then dy/dt then dy/dx = (dy/dt)/(dx/dt). For Example if x = sin t and y = cos t then, dy/dx = (dy/dt)/(dx/dt) = -sin t/cos t = tan t

  Higher Order Derivatives

  Higher Order Derivatives simply mean finding a derivative of a derivative. It is the method of finding successive differentiation.

  • If a function is differentiated two times it is called second order derivative expressed as d2y/dx2.
  • Thus in general terms, we can say that nth-order derivatives means finding successive differentiation ‘n’ times.
  • It is represented as d n y/dx n . For Example, if we have to find the derivative of a function y = 2x 3 then dy/dx = 6x 2, and its second-order derivative is d 2 y/dx 2 = d(6x 2 )/dx = 12x.

  Partial Derivative

  If we have a function given as f(x, y) then its partial derivative is given with respect to x as ∂∮f(x, y)/∂x, and its partial derivative with respect to y is given as ∂f(x, y)/∂y.

  • It should be noted that while partially differentiating a multivariable function with respect to a variable x then the variable ‘y’ in the function should be treated as constant and while partially differentiating the function with respect to y treat the variable ‘x’ as constant.
  • For Example, if we have to find the partial differentiation of f(x, y) = x 4 y 2 with respect to x and y then
  1. ∂f(x, y)/∂x = ∂(x 4 y 2 )/∂x = 4x 3 y 2
  2. ∂f(x, y)/∂y = ∂(x 4 y 2 )/∂x = 2x 4 y

  Logarithmic Differentiation

  Logarithmic Differentiation is a method of finding the differentiation of a complex function after simplifying it using Logarithm Rules.

  Let’s say we have to differentiate a function y = x x then we need to simplify it first by taking log on both sides and then differentiating it.

  Taking the natural log of the above equation, we get

  ln y = x ln x

  Now Differentiating both sides

  d(ln y)/dx = d(x ln x)/dx

  ⇒ (1/y)·dy/dx = x.1/x + ln x · 1

  ⇒ dy/dx = y(1 + ln x)

  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.
  1. If f'(x) ≥ 0 in the interval then the nature of the function is increasing in the interval.
  2. 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

  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:

  If x = a is point of local minima then

  • f'(a) = 0 and
  • f'(x) > 0 for points right to x = a and f'(x) < 0 for points left to x = a

  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

  Derivatives Examples

  Here are some examples of derivatives as illustration of the concept. In this, we will learn how to differentiate some commonly used functions such as sin x, cos x, tan x, sec x, cot x, and log x using different methods.

  Derivative of sin x

  We will find the derivative of Sin x using the First Principle.

  We have f(x) = sin x. Using First Principle, the derivative is given as

  Replacing f(x) with sin x and f(x + h) with sin(x + h) then we have

  f'(x) = lim _h→0 [sin(x + h) – sin(x)]/h

  Inside the bracket we sin(x + h) – sin(x), we can expand this using the formula sin C – sin D = 2 cos [(C + D)/2] sin [(C – D)/2]

  ⇒ f'(x) = lim _h→0 [2cos(x + h + x) sin(x + h – x)/2]/h

  ⇒ f'(x) = lim _h→0 [2cos(2x + h)/2 sin(h/2)]/h

  Using Limit Formula

  ⇒ f'(x) = lim _h→0 [2cos(2x + h)/2] limh→0 sin(h/2)]/h/2

  since, h→0, this implies h/2→0

  we know that lim _x→0 sin x /x = 1 ⇒ lim _h/2→0 sin(h/2)]/(h/2) = 1

  Hence, f'(x) = [cos (2x + 0)/2] ⨯ 1 = cos x

  Hence, the Derivative of Sin x is Cos x.

  Derivative of cos x

  We will find derivative of Cos x using the First Principle.

  We have, f(x) = cos x

  By first principle, f'(x) = lim _h→0 [f(x + h) – f(x)]/ h

  Replacing f(x) by cos x and f(x + h) by cos(x + h)

  ⇒ f'(x) = lim _h→0 [cos(x + h) – cos(x)]/ h

  Expanding cos(x + h) using cos (A + B) formula,

  we have, cos (x + h) cos x cos h – sin x sin h

  ⇒ f'(x) = lim _h→0 [cos x cos h – sin x sin h – cos x]/ h

  ⇒ f'(x) = (0) cos x – (1) sin x

  Hence, derivative of cos x is -sin x.

  Derivative of tan x

  We know that tan x = sin x / cos x. Hence we have f(x) = sin x / cos x. Assume u = sin x and v = cos x. From Quotient Rule of derivative, we have,

  d/dx = vdu – udv / v 2

  ⇒ cos x d(sin x) – sin x d(cos x) / cos 2 x

  ⇒ cos x . cos x – sin x (-sin x) / cos 2 x

  ⇒ cos 2 x + sin 2 x / cos 2 x

  ⇒ 1 / cos 2 x = sec 2 x

  Hence, derivative of tan x is sec 2 x.

  Derivative of sec x

  We know that sec x = 1/cos x = (cos x) -1

  We have function as (cos x) -1 which is in the form of f(g(x))

  Hence by using the chain rule

  we have d<(cos x) -1 >/dx = -(cos x) -2 .sin x = -sin x/ cos 2 x = -sec x . tan x

  Hence, derivative of sec x = -sec x.tan x

  Derivative of cot x

  We know that cot x = 1 / tan x = (tan x) -1

  Hence we have function = (tan x) -1 which is in the form of f(g(x)).

  Thus using the chain rule we have

  d <(tan x) -1 >/ dx = -(tan x) -2 .sec 2 x = -sec 2 x / tan 2 x = -cosec 2 x

  Hence, derivative of cot x is -cosec 2 x.

  Derivative of log _e x or ln x

  We have y = log _e x

  ⇒ e y = x

  Differentiating both sides

  ⇒ d(e y )/dx = dx/dx

  ⇒ e y .dy/dx = 1

  Putting y = log _e x in e y

  we have e log _e x . dy/dx = 1

  ⇒ x. dy/dx = 1

  ⇒ dy/dx = 1/x

  Hence, derivative of log _e x or ln x is 1/x.

  Sample Problems on Derivatives

  Here we have provided you with some solved problems on Derivatives:

  Question 1: Find the derivative of the function f(x) = x 2 at x = 0 using the First Principle.

  Solution:

  Question 2: Find the derivative of the function f(x) = x 2 at x = 2 by Limit Definition.

  Solution:

  Question 3: Find the derivative of the function f(x) = x 2 + x +1 at x = 0.

  Solution:

  Question 4: Find the derivative of the function f(x) = e x at x = 0.

  Solution:

  f'(x) = [Tex]\lim_\frac [/Tex]

  f'(x) = [Tex]\lim_\frac – e^x>[/Tex]

  ⇒ f'(x) = [Tex]\lim_\frac[/Tex]

  ⇒ f'(x) = [Tex]e^x\lim_\frac<(e^h – 1)>[/Tex]

  This is 0/0 form of the limit. We know that [Tex]\lim_\frac<(e^h – 1)> = 1[/Tex]

  ⇒ f'(x) = [Tex]e^x\lim_\frac[/Tex]

  ⇒ f'(x) = [Tex]e^x (1)[/Tex]

  ⇒f'(x) =e x

  f'(0) = 1

  Notice that the derivative of exponential function is exponential itself.

  Articles related to Derivatives:

  Practice Problems on Derivatives

  Problem 1: Find the derivative of the function [Tex]f(x) = 3x^2 + 2x – 5 [/Tex]

  Problem 2: Calculate the derivative of the function [Tex]g(x) = \sin(x) + \cos(x) [/Tex]

  Problem 3: Determine the derivative of the function [Tex]h(x) = e^ [/Tex]

  Problem 4: Find the derivative of the function [Tex]k(x) = \ln(x^2 + 1) [/Tex]

  Problem 5: Compute the derivative of the function [Tex]m(x) = \frac [/Tex]

  Summary

  Derivatives are a cornerstone of calculus, serving as a powerful tool for understanding how functions change. They measure the rate at which a function’s value changes as its input changes, essentially capturing the function’s instantaneous rate of change. By providing a way to analyze the slope of a function at any given point, derivatives allow us to determine various properties such as the function’s increasing or decreasing behavior, concavity, and local extrema.

  In practical terms, derivatives have widespread applications, from optimizing business processes and analyzing physical systems to solving complex problems in engineering and science.

  FAQs on Derivatives

  What is derivatives in maths?

  Derivative, in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations

  What is the differentiation of ln x?

  The differentiation of ln x is 1/x

  What are the derivatives of trigonometric function?

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

  What is the power rule of derivative?

  The power rule of derivatives states that if a function x is raised to some power n i.e. xn then d(x n )/dx = nx n-1 .

  What are derivatives in finance?

  Derivatives in finance are financial instruments whose value is derived from the value of an underlying asset, index, rate, or other reference. They include options, futures, forwards, and swaps. Derivatives are used for various purposes including hedging against risks, speculating on future price movements, and arbitrage opportunities.

  What is the derivative symbol?

  In mathematics, derivative is typically denoted by d/dx , which represents the rate of change of a function with respect to its independent variable x .

  Why is it called derivative?

  In mathematics, term “derivative” is used because it represents the rate at which a function changes with respect to its independent variable. Geometrically, the derivative at a certain point corresponds to the slope of the tangent line to the curve at that point. It’s called a derivative because it describes how one quantity is derived from another through the process of differentiation in calculus.