Derivatives | First and Second Order Derivatives, Formulas and Examples

A derivative is a concept in mathematics that measures how a function changes as its input changes. For example:

• If you're driving a car, the derivative of your position with respect to time is your speed. It tells you how fast your position is changing as time passes.
• If you're looking at a graph of a curve, the derivative at any point gives the slope of the tangent line to that curve at that point, indicating the rate of change.

Mathematical Definition of Derivative

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. 

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. For any function f(x), its derivative is given as:

Where,

• f′(x) represents the derivative of the function f(x) with respect to x.
• h represents the change in the x-values between the two points.
• f(x+h)−f(x) represents the change in the y-values between the two points.
• The limit as h approaches zero ensures that the secant line becomes the tangent line, providing the instantaneous rate of change of the function at the point x.

Read more about First Principles of Derivatives.

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{df(x)/dx} = d²y/dx² = f''(x)

n^th Order Derivative

n^th Order Derivative refers to finding successive differentiation of a function 'n' number of times. It is represented as d^yn/dxⁿ = fⁿ(x).

Read More about Higher Order Derivatives.

Rules of Derivatives

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

Read More about Derivative Rules.

Various Derivative Techniques

Implicit Differentiation: Implicit functions involve two or more variables and uses the chain rule to differentiate the function.

Parametric Derivative: When x = f(t) and y = g(t), and both are differentiable with respect to t. Then, parametric derivative is dy/dx = (dy/dt)/(dx/dt).

Partial Derivative: For a function f(x,y), the partial derivative with respect to x is 𝛛f(x,y)/𝛛x, and with respect to y is 𝛛f(x,y)/𝛛y

• When differentiating with respect to x, treat y as constant, and vice versa.

Logarithmic Derivative: This method simplifies the differentiation of complex functions using logarithmic rules.

Read More about Derivative Formulas.

Let's consider an real life problem and its solution as follows:

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.

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.

Read More about Applications of Derivatives.

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) = lim_h→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) = 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

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

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) = lim_h→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.

Read More,

• Derivative of tan x
• Derivative of sec x
• Derivative of cot x
• Derivative of cosec x
• Derivative of ln 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:

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

Solution:

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

Solution:

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

Solution: 

Practice Problems on Derivatives

Problem 1:Find the derivative of the function f(x) = 3x^2 + 2x - 5

Problem 2: Calculate the derivative of the function g(x) = sin(x) + cos(x)

Problem 3: Determine the derivative of the function h(x) = e^{2x}

Problem 4: Find the derivative of the function k(x) = ln(x² + 1)

Problem 5: Compute the derivative of the function m(x) = (3x+2)/(x-1)

Suggested Quiz

10 Questions

Which of the following is the derivative of f(x) = √x?

• A

  1/2√x

• B

  2√x

• C

  √x/2

• D

  -1/2√x

Explanation:

As we know, f'(x) = √x = x^1/2

Using the power rule, we get

f'(x) = (1/2)x^1/2-1

⇒ f'(x) = (1/2)x^-1/2

⇒ f'(x) = 1/2x^1/2

⇒ f'(x) = 1/2√x

What is the derivative of f(x) = ln⁡(x² + 1)?

• A

  [Tex]\frac{2x}{x^2 + 1}[/Tex]

• B

  [Tex]\frac{x}{x^2 + 1}[/Tex]

• C

  [Tex]\frac{2x}{\ln(x^2 + 1)}[/Tex]

• D

  [Tex]\frac{2}{x^2 + 1}[/Tex]

Explanation:

f(x) = ln⁡(x² + 1)

⇒ f'(x) = 1/(x² + 1) d/dx(x² + 1) [As d/dx(ln x) = 1/x]

⇒ f'(x) = 2x/(x² + 1)

Differentiate [Tex]f(x) = \ln\left(\frac{x}{x+1}\right)[/Tex].

• A

  1/x − 1/(x + 1)

• B

  1/x + 1/(x + 1)

• C

  -1/x + 1/(x + 1)

• D

  1/x(x + 1)

Explanation:

[Tex]f(x) = \ln\left(\frac{x}{x+1}\right)[/Tex]

⇒ [Tex]f(x) = \ln x - \ln (x+1)[/Tex]

⇒ f'(x) = 1/x - 1/(x + 1)

If [Tex]y=1+x+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{3}}}{3\,!}+\cdots +\frac{{{x}^{n}}}{n\,!}[/Tex] then dy/dx =

• A

  y

• B

  y + xⁿ/n!

• C

  y - xⁿ/n!

• D

  y - 1 - xⁿ/n!

Explanation:

Given: [Tex]y=1+x+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{3}}}{3\,!}+.....+\frac{{{x}^{n}}}{n\,!}[/Tex]

⇒ [Tex]\frac{dy}{dx} = 0 + 1+\frac{2x}{2!}+\frac{3x^{2}}{3!}+ \cdots + \frac{nx^{n-1}}{n!}[/Tex]

⇒ [Tex]\frac{dy}{dx} = 1+ x +\frac{x^{2}}{2!}+ \cdots + \frac{x^{n-1}}{(n-1)!}[/Tex]

⇒[Tex] \frac{dy}{dx} = 1+ x +\frac{x^{2}}{2!}+ \cdots + \frac{x^{n-1}}{(n-1)!} + \frac{x^{n}}{n!} - \frac{x^{n}}{n!}[/Tex]

⇒ [Tex]\frac{dy}{dx} = y - \frac{x^{n}}{n!}[/Tex]

d/dx [log|x|] = ___ (x ≠ 0)

• A

  1/x

• B

  -1/x

• C

  [Tex]\begin{cases} 1/x,& x>0\\-1/x, & x<0\end{cases}[/Tex]

• D

  None of These

Explanation:

[Tex]\frac{d}{dx} \log |x| = \frac{1}{|x|} \cdot \frac{d}{dx} |x| = \frac{1}{|x|} \cdot \frac{x}{|x|} [/Tex]

⇒ [Tex]\frac{d}{dx} \log |x| = \frac{x}{x^2} = \frac{1}{x}[/Tex]

If y = a sinx + b cosx, then y² + (dy/dx)² is

• A

  function of x

• B

  function of y

• C

  function of x and y

• D

  constant

Explanation:

Given: y = a sinx + b cosx

⇒ [Tex]y^2 = (a \sin x + b \cos x)^2[/Tex]

⇒ [Tex]y^2 = a^2 \sin^2 x + b^2 \cos^2 x + 2ab \sin x \cos x [/Tex]

and [Tex]\frac{dy}{dx} = a \cos x - b \sin x[/Tex]

⇒ [Tex]\left(\frac{dy}{dx}\right)^2 = (a \cos x - b \sin x)^2[/Tex]

⇒ [Tex]\left(\frac{dy}{dx}\right)^2 = a^2 \cos^2 x + b^2 \sin^2 x - 2ab \sin x \cos x [/Tex]

Now, [Tex]y^2 + \left(\frac{dy}{dx}\right)^2 = \left(a^2 \sin^2 x + 2ab \sin x \cos x + b^2 \cos^2 x\right) + \left(a^2 \cos^2 x - 2ab \sin x \cos x + b^2 \sin^2 x\right)[/Tex]

⇒ [Tex]y^2 + \left(\frac{dy}{dx}\right)^2 = a^2 \sin^2 x + b^2 \cos^2 x + a^2 \cos^2 x + b^2 \sin^2 x[/Tex]

⇒ [Tex]y^2 + \left(\frac{dy}{dx}\right)^2 = a^2 (\sin^2 x + \cos^2 x) + b^2 (\sin^2 x + \cos^2 x)[/Tex]

⇒ [Tex]y^2 + \left(\frac{dy}{dx}\right)^2 = a^2 + b^2[/Tex]

If y=\sqrt{(1-x)(1+x)}, then

• A

  (1-{{x}^{2}})\frac{dy}{dx}-xy=0

• B

  (1-{{x}^{2}})\frac{dy}{dx}+xy=0

• C

  (1-{{x}^{2}})\frac{dy}{dx}-2xy=0

• D

  (1-{{x}^{2}})\frac{dy}{dx}+2xy=0

Explanation:

Given: [Tex]y=\sqrt{(1-x)(1+x)}[/Tex]

⇒ [Tex]y = \sqrt{1 - x^2}[/Tex]

⇒ [Tex]\frac{dy}{dx} = (1-x^2)^{-1/2} \frac{d}{dx}(1-x^2)[/Tex]

⇒ [Tex]\frac{dy}{dx} = \frac{1}{1-x^2)^{1/2}} \times (-2x)[/Tex]

⇒ [Tex]\frac{dy}{dx} = \frac{-2x}{\sqrt{1-x^2}}[/Tex]

⇒ [Tex](1-x^2)\frac{dy}{dx} = \frac{-2x}{\sqrt{1-x^2}} \times (1-x^2)[/Tex]

⇒ [Tex](1-x^2)\frac{dy}{dx} = -2x \times (\sqrt{1-x^2})[/Tex]

⇒ [Tex](1-x^2)\frac{dy}{dx} = -2xy[/Tex]

⇒ [Tex](1-x^2)\frac{dy}{dx} + 2xy = 0[/Tex]

For a function [Tex]y = f(x)={{\log }_{10}}x+{{\log }_{x}}10+{{\log }_{x}}x+{{\log }_{10}}10[/Tex], What is the first-order derivative?

• A

  [Tex]\frac{1}{x{\log _e}10}-\frac{ \log_{e}10}{x({\log }_{e}x)^{2}}[/Tex]

• B

  [Tex]\frac{1}{x{\log _e}10}-\frac{ \log_{e}10}{x{\log }_{e}x}[/Tex]

• C

  [Tex]\frac{1}{x{\log _e}10}+\frac{ \log_{e}10}{x({\log }_{e}x)^{2}}[/Tex]

• D

  None of These

Explanation:

Using log_ba = ln a/ln b and log_aa = 1, given expression simplifies to:

[Tex]f(x)= \frac{\ln x}{\ln 10} + \frac{\ln 10}{\ln x} + 1 + 1[/Tex]

⇒ [Tex]f'(x)= \frac{1}{x\cdot \ln 10} + \frac{\ln 10}{x\cdot (\ln x)^2} + 0[/Tex]

⇒ [Tex]f'(x)= \frac{1}{x \ln 10} + \frac{\ln 10}{x (\ln x)^2}[/Tex]

[Tex]\frac{d}{dx}{{\log }_{7}}({\log }_{7}x) = [/Tex]

• A

  [Tex] \frac{\log_x 7}{x(\ln 7)^2}[/Tex]

• B

  [Tex] \frac{1}{x\log_7 x \ln 7}[/Tex]

• C

  [Tex] \frac{1}{x\ln x (\ln 7)^2}[/Tex]

• D

  None of These

Explanation:

As we know, \frac{d}{dx} \log_b x = \frac{1}{x \ln b}

[Tex]\frac{d}{dx} \log_7 (\log_7 x) = \frac{1}{\log_7 x \ln 7} \frac{d}{dx}(\log_7 x) = \frac{1}{\log_7 x \ln 7} \cdot \frac{1}{x \ln 7}[/Tex]

⇒ [Tex]\frac{d}{dx} \log_7 (\log_7 x) = \frac{1}{x\log_7 x (\ln 7)^2} = \frac{\log_x 7}{x(\ln 7)^2}[/Tex]

[Tex]\frac{d}{dx} \left(e^{x+3\log x}\right) =[/Tex]

• A

  [Tex]e^x\cdot x^2(x + 3)[/Tex]

• B

  [Tex]e^x\cdot x(x + 3)[/Tex]

• C

  [Tex]e^x + \frac{3}{x}[/Tex]

• D

  None of These

Explanation:

[Tex]e^{x+3\log x} = e^x \cdot e^{3\log x} = e^x \cdot e^{\log x^3} = e^x \cdot x^3[/Tex] [As e^{log x} = x]

[Tex]\frac{d}{dx} \left(e^{x+3\log x}\right) = \frac{d}{dx}\left(e^x \cdot x^3\right)[/Tex]

⇒ [Tex]\frac{d}{dx}\left(e^x \cdot x^3\right) = x^3 \cdot e^x + 3x^2 \cdot e^x = e^x\cdot x^2(x + 3)[/Tex]

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