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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.

Derivatives form an important part of the Limits and Differentiation chapter in NCERT class 11 and 12 textbooks. Let’s learn about Derivatives in detail, including their types, examples, formulas, and applications.

Derivatives Meaning

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, 

[Tex]\bold{f'(a) = \lim_{h \to 0} \frac{f(a + h) -f(a)}{h}}[/Tex]

The above statement is subject to the condition that its limits exist.  This is also referred to as [Tex]\left.\dfrac{df}{dx}\right|_{x = a}[/Tex]

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:

Derivative by First Principle

  • 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, 

[Tex]\bold{f'(a) = \lim_{h \to 0} \frac{f(a + h) -f(a)}{h}}[/Tex]

  • 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, 

[Tex]\bold{\lim_{h \to 0} \frac{f(a + h) -f(a)}{h} = \lim_{Q \to P}\frac{QP}{QR}}[/Tex]

  • 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.

Derivatives Types

The different types of Derivatives include,

  • First Order Derivative,
  • Second Order Derivative, and
  • 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) = limx→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{df(x)/dx} = d2y/dx2 = f”(x)

nth Order Derivative

nth Order Derivative refers to finding successive differentiation of a function ‘n’ number of times. It is represented as dyn/dxn = fn(x).

Derivatives Formulas

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) = xn then its derivative

dy/dx = f'(x) = nxn-1

where n is an integer

For example, the derivative of f(x) = x3 is 3x(3-1) = 3x2.

Derivative of Exponential Function

The derivative of the Exponential Function is listed below:

  • d(ex)/dx = ex
  • d(ax) = ax ln a

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

Derivative of Logarithmic Function

The formula for derivatives of logarithmic functions is given below:

  • d(ln x)/dx = 1/x
  • d(logax)/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 = sec2x
  • d(cot x)/dx = -cosec2x
  • 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-1x is the inverse trigonometric function.

The derivative formula for inverse trigonometric function is given below:

  • d(sin-1x)/dx = 1/√(1 – x2)
  • d(cos-1x)/dx = -1/√(1 – x2)
  • d(tan-1x)/dx = 1/(1 + x2)
  • d(cosec-1x)/dx = -1/|x|√(x2 – 1)
  • d(sec-1x)/dx = 1/|x|√(x2 – 1)
  • d(cot-1x)/dx = -1/(1 + x2)

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

Following is the Power Rule of Derivatives:

  • If f(x) = xn then d(f(x))/dx = dxn/dx = nxn-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

d(c.f(x))/dx = c.f'(x)

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 

d(u.v)/dx = u.dv/dx + v.du/dx

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)v2

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 = sin2x 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 = u2 where u = sin x

Step 2: Now derivative of u = sin x with respect to x and y = u2 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 sin2x 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) = sin2x then f'(x) = 2sin x cos x. In this example we first differentiated sin2x 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(3xy2)/dx = 3y2 + 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

d(3x + y)/dx = d(5)/dx

⇒ 3 + dy/dx = 0

⇒ dy/dx = -3

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 dny/dxn. For Example, if we have to find the derivative of a function y = 2x3 then dy/dx = 6x2, and its second-order derivative is d2y/dx2 = d(6x2)/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) = x4y2 with respect to x and y then
  1. ∂f(x, y)/∂x = ∂(x4y2)/∂x = 4x3y2
  2. ∂f(x, y)/∂y = ∂(x4y2)/∂x = 2x4y

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 = xx then we need to simplify it first by taking log on both sides and then differentiating it.

y = xx

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.

The condition for these are tabulated below:

Maxima and Minima

Absolute Maximumx = af(x) ≤ f(a) where x and a belong to domain of f.
Absolute Minimumx = af(x) ≥ f(a) where x and a belong to  domain of f.
Local Maximumx = af(x) ≤ f(a) for all x near to a.
Local Minimumx = af(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

  • 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

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

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) < 0x = a is point of local maxima
f”(a) > 0x = a is point of local minima
f”(a) = 0x = a may be a point of local maxima or local minima or none.

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

f'(x) = limh→0 [f(x + h) – f(x)]/ h

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

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

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

Using Limit Formula

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

since, h→0, this implies h/2→0

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

we know that limx→0 sin x /x = 1 ⇒ limh/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) = limh→0 [f(x + h) – f(x)]/ h

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

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

⇒ f'(x) = limh→0 {(cos h – 1)/h}cos x – {sin h/h}sin x

⇒ f'(x) = (0) cos x – (1) sin x

⇒ f'(x) = -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{u/v}/dx = vdu – udv / v2

⇒ cos x d(sin x) – sin x d(cos x) / cos2x

⇒ cos x . cos x – sin x (-sin x) / cos2x

⇒ cos2x + sin2x / cos2x

⇒ 1 / cos2x = sec2x

Hence, derivative of tan x is sec2x.

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/ cos2 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.sec2x = -sec2x / tan2x = -cosec2x

Hence, derivative of cot x is -cosec2x.

Derivative of logex or ln x

We have y = logex

⇒ ey = x

Differentiating both sides

⇒ d(ey)/dx = dx/dx

⇒ ey.dy/dx = 1

Putting y = logex in ey

we have elogex . dy/dx = 1

⇒ x. dy/dx = 1

⇒ dy/dx = 1/x

Hence, derivative of logex 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) = x2 at x = 0 using the First Principle.

Solution:

f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h}       [/Tex] 

f'(x) = [Tex]\lim_{h \to 0}\frac{(x + h)^2 – x^2}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{(x^2 + h^2 + 2hx – x^2)}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}h + 2x[/Tex]

⇒ f'(x) = 2x

f'(0) = 0

Question 2: Find the derivative of the function f(x) = x2 at x = 2 by Limit Definition.

Solution:

f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h}       [/Tex] 

f'(x) = [Tex]\lim_{h \to 0}\frac{(x + h)^2 – x^2}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{(x^2 + h^2 + 2hx – x^2)}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}h + 2x[/Tex]

⇒ f'(x) = 2x

f'(2) = 4

Question 3: Find the derivative of the function f(x) = x2 + x +1 at x = 0. 

Solution:

f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h}       [/Tex] 

f'(x) = [Tex]\lim_{h \to 0}\frac{((x + h)^2 + (x +h) + 1) – (x^2 + x + 1)}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{((x^2 + h^2 + 2hx + (x +h) + 1) – (x^2 + x + 1)}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{(h^2 + 2hx +h)}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}(h + 2x +1)[/Tex]

⇒f'(x) = 2x + 1

f'(0) = 1 

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

Solution: 

f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h}       [/Tex] 

f'(x) = [Tex]\lim_{h \to 0}\frac{e^{(x + h)} – e^x}{h}[/Tex]

⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{e^xe^h – e^x}{h}[/Tex]

⇒ f'(x) = [Tex]e^x\lim_{h \to 0}\frac{(e^h – 1)}{h}[/Tex]

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

⇒ f'(x) = [Tex]e^x\lim_{h \to 0}\frac{e^h }{1}[/Tex]

⇒ f'(x) = [Tex]e^x (1)[/Tex]

⇒f'(x) =ex

f'(0) = 1 

Notice that the derivative of exponential function is exponential itself. 

Related Articles

Differentiation Formulas

Product Rule of Derivatives

Quotient (Division) Rule of Derivatives

Increasing and Decreasing Functions

Tangent and Normal

Applications of Derivatives

Antiderivatives

Differentiation and Integration Formula

FAQs on Derivatives

What is derivative in Maths?

Derivative in Maths is the rate of the instantaneous change in a function with respect to another independent variable.

What is the differentiation of ln x?

The differentiation of ln x is 1/x

What are the derivatives of trigonometric function?

The derivatives of trigonometric functions are listed below:

  • 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(xn)/dx = nxn-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.



Last Updated : 29 Feb, 2024
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