Derivative of a Function is the rate of change in the given function with respect to an independent variable. The derivative of a given function in Calculus is found using the First Principle of differentiation. The process of finding the derivative of a function is also called the Differentiation of function.

Derivative of a Function gives the slope of the particular function at the point of differentiation and is used to get the extreme value of the function. It is also defined as the rate of change of the function with respect to any point lying in the domain of the function. A function f(x) is differentiable at a point x = a if it is continuous at that point.

In this article, we will learn about the Derivative of a function, the First Principle of Differentiation, the Differentiation of Trig Function, the Differentiation of Exponential Function, and others in detail.

What is Derivative of A Function

Derivative of a Function is the slope of the function at any particular point in the domain of the function. We know that every function represents a curve in space and the derivative of the function at any particular point represents the slope of the particular function at that point. For any function f(x) is said to be differentiable if the differentiation of the function exists and it is denoted by f'(x) or df(x)/dx. We also represent a function as, y = f(x) then

Derivative of a Function is represented as f'(x) or y’ or dy/dx

It is called the derivative of y with respect to x.

Derivative of a function is also define as the rate of change of the function with respect to any point on its domain.

Read More:

• Calculus in Maths
• How to Find Derivative of a Function
• Algebra of Derivative of Functions

First Principle of Differentiation

Suppose a function f(x) is defined in the neighborhood of ‘a’. If a and a + h belong to the domain of function f(x) then the differentiation of the function at x = a is given by,

f'(a) = lim _x→a {f(x + h) – f(x)}/h

This is known as the first principle of differentiation. We use this first principle to find the derivative of the function at any given point and this derivative gives the slope of the function at that particular point.

Note: If the derivative of f(x) exists then the function f is said to be derivable or differentiable. Differentiation is the process of finding derivatives.

Geometrical Interpretation of Derivative of a Function

Consider the graph added below,

Function f(x) defined on an open interval (a, b). Let P be a point on the curve y = f(x). Let Q[(c – h), f(c – h)] and R be the point on either side of point P. Now,

Slope of chord PQ is, {f(c – h) – f(c)} / (-h)

Slope of chord PR can be written as, {f(c + h) – f(c)} / h

Now, we know that tangent to a curve at a point P is the limiting position of secant PQ when Q tends to P.

Similarly, it is also the limiting position of secant PR when R tends to P.

∴ h⇢ 0 points Q and R both tend to P from left and right hand sides.

∴ Slope of tangent at point P is,

[Tex]P = \lim \;_{Q \to P} \;(Slope\ of\ secant\ PQ) [/Tex]

[Tex]\lim \;_{h \to 0} \;(Slope\ of\ secant\ PQ) [/Tex]

[Tex]\lim \;_{Q \to P} \; \frac{f(c + h) – f(c)}{h} [/Tex]

If these limits exist and are equal, there is a unique tangent at point P.

Slope of tangent is denoted by dy/dx i.e., f'(x)

Thus,

f'(x) = lim _x→a {f(x + h) – f(x)}/h = lim _x→a {f(x + h) – f(x)}/{-h}

How to Find Derivative of Function?

To find the derivative of a function we use the first principle formula, i.e. for any given function f(x) whose derivative at x = a is to be found the first principle formula is,

f'(x) = lim _x→a {f(x + h) – f(x)}/h

Simplifying the above we get the required derivative of the function at any point in the domain of the function.

Derivative of a Standard Functions

Derivative of some standard function are given below that are used to solve various problems of the differentiation. That includes,

Derivative of Constant Function

The constant functions are the functions whose value does not change with respect to the change in the value of the parameter of the function. They generally represent a straight line in the graph of the function. Derivative of the constant function is found as,

Let f(x) = k {where k is any constant}

∴ f(x + h) = k

Consider,

f'(x) = lim _h→0 {f(x + h) – f(x)}/h

f'(x) = lim _h→0 {k – k}/h = 0

f'(x) = 0

d(k)/dx = 0

Derivative of Exponential Function

Exponnetial Function are the function that are in the form of e^x and the differentiation of the function is found using the first principle as,

Let f(x) = e^x

f(x + h) = e^(x+h)

f'(x) = lim _h→0 {f(x + h) – f(x)}/h

f'(x) = lim _h→0 {e^(x+h) – e^x}/h

f'(x) = lim _h→0 e^x{e^h – 1}/h

f'(x) = e^x

Derivative of Trigonometric Functions

Trigonometric functions are the function that uses trigonometric values and their differentiation is also easily found using the first principle concept as, suppose we have to find the differentiation of sin x then,

Let f(x) = sin x

f(x + h) = sin(x + h)

f'(x) = lim _h→0 {f(x + h) – f(x)}/h

f'(x) = lim _h→0 {sin (x + h) – sin (x)}/h

f'(x) = lim _h→0 {2 cos (x + h + x)/2 . sin (x + h – x)/2}/h

f'(x) = lim _h→0 {cos (2x + h)/2 . sin (h/2)}/(h/2)

f'(x) = lim _h→0 {cos (2x + h)/2} . lim _h/2→0 sin (h/2)/(h/2)

f'(x) = cos {(2x +0)/2}.1

f'(x) = cos (2x/2).1 = cos x

Thus, the derivative of sin x is cos x proved.

Similarly derivative of other trigonometric functions are added below,

• d(cos x)/dx = – sin x
• d(tan x)/dx = sec² x
• d(cot x)/dx = – cosec² x
• d(sec x)/dx = sec x . tan x
• d(cosec x)/dx = – cosec x . cot x

Derivative of Absolute Function

Absolute function is the function that gives the distance of point from origin. It is denoted using |x| and its derivative is calculated as,

d/dx(|x|) = x/|x|

where, x ≠ 0

Derivative of Logarithmic Function

Logarithimic Function are also called the log function and is represented as, log x and its derivative is found as,

d/dx (log x) = 1/x

Derivative of Inverse Function Formula

For any function f(x) its inverse is defined as f^-1(x) and the derivative of the inverse function is found using the formula,

d/dx[f^-1(x)] = 1/f'{f^-1(x)}

Derivative of Implicit Function

Implict functions are the functions that can not be easily solve for y and their differention is called the Implict Differentiation. Derivative of Implict function is found using the chain rule.

Learn more about, Implicit Differentiation

Derivative of Composite Function

For any composite function h(x) such that, h(x) = f{g(x)}, to find the derivative of the composite function h(x) we first find the derivative of the composite function and then multiply the same with the derivative of g(x), i.e.

d/dx.f{g(x)} = f'(g(x)).g'(x)

Learn more about, Derivatives of Composite Functions

Algebra of Derivatives

We can also very easily perform algebra on the derivative of the functions. Suppose we are given with two differentiable function f(x) and g(x) then the rules of algebra state that,

• Sum of derivatives of the functions f and g is equal to the derivative of their sum, i.e.,

d/dx {f(x) + g(x)} = d/dx.f(x) + d/dx.g(x)

• Difference of derivatives of the functions f and g is equal to the derivative of their difference, i.e.,

d/dx {f(x) – g(x)} = d/dx.f(x) – d/dx.g(x)

• If we are given two function as the product then the derivative of the function is calculated as,

d/dx{f(x).g(x)} = f(x).d/dx{g(x)} + g(x).d/dx{f(x)}

This is also called the product rule in calculus.

• If we are given two function in quotient form then the derivative of the function is calculated as,

d/dx{f(x)/g(x)} = [f(x).d/dx{g(x)} – g(x).d/dx{f(x)}]/{g(x)}²

Derivative Formulas

Various Derivative Formulas are,

• (d/dx) constant = 0
• (d/dx) xⁿ = nx^n-1
• d/dx) e^x = e^x
• (d/dx) a^x = a^x ln a
• (d/dx) ln x = (1/x)

Apart form these formulas some other derivative formulas are,

• (d/dx) [f(x) ± g(x)] = (d/dx) f(x) ± (d/dx) g(x)
• (d/dx) [f(x). g(x)] = f'(x). g(x) + f(x). g'(x)
• (d/dx) [f(x)/g(x)] = [f'(x). g(x) – f(x). g'(x)]/[g(x)]²
• (d/dx) [f(g(x))] = (d/dx) [f(g(x))] × (d/dx) [g(x)]

Learn more about, Derivative Formulas

Read More,

• Differentiation Formulas
• Logarithmetic Differentiation

Examples on Derivative Rules

Example 1: Find the derivative of 3x + 4 using the first principle of derivative.

Solution:

Let f(x) = 3x + 4

f(x + h) = 3(x + h) + 4

Then,

[Tex]\lim\;_{h \to 0} \; \frac{f(x + h) – f(x)}{h} [/Tex]

[Tex]\lim\;_{h \to 0} \; \frac{3(x + h)\; + \; 4 \; – \;(3x + 4)}{h} [/Tex]

[Tex]\lim\;_{h \to 0}\; \frac{3h}{h} [/Tex]

= 3

∴ f’ (x) = 3

Example 2: Find the derivative of

• 1 / x²
• x cos x

Using the First Principle of Derivative.

Solution:

• 1 / x²

Let f(x) = 1 / x²

f(x + h) = 1 / (x + h)²

Then,

[Tex]\lim\;_{h \to 0} \; \frac{f(x + h) – f(x)}{h} [/Tex]

[Tex]\lim\;_{h \to 0} \; \frac{\frac{1}{(x+h)^2} – \frac{1}{x^2}}{h} [/Tex]

[Tex]\lim\;_{h \to 0}\; \frac{x^2 – (x + h^2)}{h(x + h)^2 . x^2} [/Tex]

[Tex]\lim\;_{h \to 0}\; \frac{x^2 – (x^2 + 2xh + h^2)}{h(x + h)^2 . x^2} [/Tex]

[Tex]\lim\;_{h \to 0}\; \frac{-h(2x + h)}{h(x + h)^2 . x^2} [/Tex]

[Tex]-\lim\;_{h \to 0}\; (2x + h)\;\;\lim\;_{h \to 0}\; \frac{1}{(x + h)^2 . x^2} [/Tex]

= -2x(1 / x²x²)

= -2 / x³

∴ f'(x) = -2 / x³

• x cos x

Let f(x) = x cos x

f(x + h) = (x + h) cos (x + h)

consider,

[Tex]\lim\;_{h \to 0} \; \frac{f(x + h) – f(x)}{h} [/Tex]

[Tex]\lim\;_{h \to 0} \; \frac{(x + h) cos (x + h) – x cos x}{h} [/Tex]

[Tex]\lim\;_{h \to 0} \; \frac{x[cos (x + h) – cos x]+ h\;cos(x + h)}{h} [/Tex]

[Tex]\lim\;_{h \to 0} \; \frac{x(-2)sin(x + \frac{h}{2})\;sin\frac{h}{2}}{2x\frac{h}{2}}\; + \; \lim\;_{h \to 0}\; \frac{h\;cos(x + h)}{h} [/Tex]

[Tex]-x\;\lim\;_{h \to 0}\; sin (x +\frac{h}{2}).\; \lim\;_{h \to 0}\; \frac{sin\frac{h}{2}}{\frac{h}{2}}\;+\; cos(x + 0) [/Tex]

= -x sin x. 1 + cos x

= -x sin x + cos x

∴ f'(x) = -x sin x + cos x

Example 3: Differentiate the following functions,

• sin x / 1 + sin x
• e^x / 1 + sin x

Solution:

• sin x / 1 + sin x

we have

y = sin x / 1 + sin x

[Tex]\frac{dy}{dx} = \frac{(1 + sin\;x) \frac{d}{dx}(sin\;x) – sin\;x \frac{d}{dx}(1 + sin\;x)}{(1+ sinx)^2} [/Tex]

[Tex]=\frac{(1 + sin\;x).cos\;x  – sin\;x. (0 + cos\;x)}{(1+ sinx)^2} [/Tex]

[Tex]=\frac{cos\;x + sin\;x.cos\;x – sin\;x.cos\;x}{(1 + sin\;x)^2} [/Tex]

= cos x / (1 + sin x)²

• e^x / 1 + sin x

y = e^x / 1 + sin x

[Tex]\frac{dy}{dx}\; = \frac{(1 + sin\;x)\frac{d}{dx}e^x\;-\;e^x\;\frac{d}{dx}(1 + sin\;x)}{(1 + sin\;x)^2} [/Tex]

[Tex]=\frac{(1 + sin\;x).e^x\;-\;e^x.(0 + cos\;x)}{(1 + sin\;x)^2} [/Tex]

[Tex]= \frac{e^x[1 + sin\;x – cos\;x]}{(1 + sin\;x)^2} [/Tex]

Example 4: y = log ₅(log₇ x) find dy / dx

Solution:

Given,

• y  = log₅ (log₇x)

Using logarithmic property, we can write,

y = log( log₇x ) / log 5

[Tex]\frac{dy}{dx} = \frac{1}{log\;5}.\frac{d}{dx}[log(log_7x)] [/Tex]

[Tex]= \frac{1}{log\;5}\;.\;\frac{1}{\frac{log\;x}{log\;7}}\;.\;\frac{d}{dx}(\frac{log\;x}{log\;7}) [/Tex]

[Tex]= \frac{1}{log\;5}\;.\;\frac{1}{\frac{log\;x}{log\;7}}\;.\;\frac{1}{log\;7}\frac{d}{dx}(log\;x) [/Tex]

[Tex]= \frac{1}{log\;5\;.\;.log\;x}\;.\;\frac{1}{x} [/Tex]

= 1 / x log 5. log x

Example 5: If y = log(3x² + 2x +1), find dy/dx

Solution:

y = log (3x² + 2x +1)

[Tex]= \frac{dy}{dx} = \frac{1}{3x^2 + 2x + 1}\;\;\frac{d}{dx}(3x^2 + 2x + 1) [/Tex]

[Tex]= \frac{1}{3x^2+2x+1}\;\;(6x +2) [/Tex]

[Tex]= \frac{2(3x +1)}{3x^2 + 2x +1} [/Tex]

Example 6: Using First Principle find the derivative of √sin x

Solution:

Given,

• f(x) = √sin x

f(x + h) = √sin (x + h)

f'(x) = lim_x→0 {f(x + h) – f(x)}/h

[Tex]f'(x)~=~\lim\;_{h \to 0}\;\frac{\sqrt{sin(x + h)} \;-\;\sqrt{sin\;x} }{h} [/Tex]

[Tex]f'(x)~=~\lim\;_{h \to 0}\;\frac{\sqrt{sin(x + h)} \;-\;\sqrt{sin\;x} }{h} \; \times \;= \frac{\sqrt{sin(x + h)} \;+\;\sqrt{sin\;x} }{\sqrt{sin(x + h)} \;+\;\sqrt{sin\;x}} [/Tex]

[Tex]f'(x)~=~\;\lim\;_{h \to 0}\; \frac{sin(x + h) – sin\;x}{h[\sqrt{sin(x + h)} \;+\; \sqrt{sin\;x} ]} [/Tex]

[Tex]f'(x)~=~\lim\;_{h \to 0}\; \frac{2 cos\;(\frac{x + h+ x}{2})\;.\;sin\;(\frac{x + h – x}{2})}{h[\sqrt{sin(x + h)}\; +\;\sqrt{sin\;x} ]} [/Tex]

[Tex]f'(x)~=~\lim\;_{h \to 0}\; \frac{2\;cos\;(x +\frac{h}{2})\;.\;sin(\frac{h}{2})}{h[\sqrt{sin(x + h)\;+\;\sqrt{sin\;x}}]} [/Tex]

[Tex]f'(x)~=~\lim\;_{h \to 0}\; \frac{\;cos\;(x +\frac{h}{2})}{\sqrt{sin(x + h)\;+\;\sqrt{sin\;x}}}\;\times\;\frac{sin(\frac{h}{2})}{(h/2)} [/Tex]

[Tex]f'(x)~=~\frac{\lim\;_{h \to 0}\;cos\;(x +\frac{h}{2})}{\lim\;_{h\to 0}\;\sqrt{sin(x + h)\;\;+\sqrt{sin\;x}}}\;\times\lim~_{h \to 0}\;\frac{sin(\frac{h}{2})}{(h / 2)} [/Tex]

[Tex]f'(x)~=~\frac{cos\;x}{2\sqrt{sin\;x}} [/Tex]

Important Maths Related Links:

• Volume of a Pyramid Formula
• What is a Pyramid
• Surface Area of a Pyramid Formula
• Square Pyramid Formula
• Pentagonal Pyramid
• Hexagonal Pyramid Formula

Practice problems: Derivative of a Function

1. Find the derivative of the function [Tex]f(x) = x^3 + 2x^2 – 5x + 7[/Tex].

2. Determine the derivative of the function g(x)=sin⁡(x)+cos⁡(x).

3. Compute the derivative of the function h(x)=e^x⋅ln⁡(x).

4. Find the derivative of the function [Tex]p(x) = \frac{x^2 + 1}{x – 1}[/Tex]​.

5. Find the derivative of the function [Tex]p(x) = \frac{x^3 + 1}{x – 1}[/Tex]​.

6. Determine the derivative of the function [Tex]q(x) = \sqrt{x^2 + 4}[/Tex].

Conclusion

The derivative of a function is a fundamental concept in calculus that measures how a function’s output changes as its input changes. It is often described as the instantaneous rate of change of the function at a given point. Mathematically, the derivative of a function f(x) at a point x=a is the limit of the average rate of change of the function over an interval as the interval approaches zero. This is expressed as[Tex] f'(a) = \lim_{\Delta x \to 0} \frac{f(a + \Delta x) – f(a)}{\Delta x}[/Tex]​. The derivative provides essential information about the behavior of functions, such as determining local maxima and minima, analyzing motion, and solving differential equations. Understanding the derivative is crucial for various applications in physics, engineering, economics, and beyond, making it a cornerstone of mathematical analysis and applied mathematics.

Derivative of a Function – FAQs

What is Derivative of a Function?

The derivative of a function is the rate of change of the function with respect to any value from the domain of the function. It gives the slope of the function at the given point.

What is Dervivative of Sin x?

The derivative of sin x is calculated as, suppose y = sin x, then

y'(x) = cos x

What is Derivative of Tan x?

The derivative of tan x is calculated as, suppose y = tan x, then

y'(x) = sec² x

What is Derivative of Cos x?

The derivative of cos x is calculated as, suppose y = cos x, then

y'(x) = -sin x

What is Derivative of e^x?

The derivative of e^x is calculated as, suppose y = e^x, then

y'(x) = e^x