Derivative of a Function

Last Updated : 18 Oct, 2023

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.

Table of Content

What is Derivative of A Function
How to Find Derivative of Function
Derivative of a Standard Functions
Derivative of Constant Function
Derivative of Exponential Function
Derivative of Trigonometric Functions
Derivative of Composite Function

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.

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,

Derivative of a Function

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,

$P = \lim \;_{Q \to P} \;(Slope\ of\ secant\ PQ)$

$\lim \;_{h \to 0} \;(Slope\ of\ secant\ PQ)$

$\lim \;_{Q \to P} \; \frac{f(c + h) - f(c)}{h}$

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^xand 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,

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,

$\lim\;_{h \to 0} \; \frac{f(x + h) - f(x)}{h}$

$\lim\;_{h \to 0} \; \frac{3(x + h)\; + \; 4 \; - \;(3x + 4)}{h}$

$\lim\;_{h \to 0}\; \frac{3h}{h}$

= 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,

$\lim\;_{h \to 0} \; \frac{f(x + h) - f(x)}{h}$

$\lim\;_{h \to 0} \; \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}$

$\lim\;_{h \to 0}\; \frac{x^2 - (x + h^2)}{h(x + h)^2 . x^2}$

$\lim\;_{h \to 0}\; \frac{x^2 - (x^2 + 2xh + h^2)}{h(x + h)^2 . x^2}$

$\lim\;_{h \to 0}\; \frac{-h(2x + h)}{h(x + h)^2 . x^2}$

$-\lim\;_{h \to 0}\; (2x + h)\;\;\lim\;_{h \to 0}\; \frac{1}{(x + h)^2 . x^2}$

= -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,

$\lim\;_{h \to 0} \; \frac{f(x + h) - f(x)}{h}$

$\lim\;_{h \to 0} \; \frac{(x + h) cos (x + h) - x cos x}{h}$

$\lim\;_{h \to 0} \; \frac{x[cos (x + h) - cos x]+ h\;cos(x + h)}{h}$

$\lim\;_{h \to 0} \; \frac{x(-2)sin(x + \frac{h}{2})\;sin\frac{h}{2}}{2x\frac{h}{2}}\; + \; \lim\;_{h \to 0}\; \frac{h\;cos(x + h)}{h}$

$-x\;\lim\;_{h \to 0}\; sin (x +\frac{h}{2}).\; \lim\;_{h \to 0}\; \frac{sin\frac{h}{2}}{\frac{h}{2}}\;+\; cos(x + 0)$

= -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

$\frac{dy}{dx} = \frac{(1 + sin\;x) \frac{d}{dx}(sin\;x) - sin\;x \frac{d}{dx}(1 + sin\;x)}{(1+ sinx)^2}$

$=\frac{(1 + sin\;x).cos\;x - sin\;x. (0 + cos\;x)}{(1+ sinx)^2}$

$=\frac{cos\;x + sin\;x.cos\;x - sin\;x.cos\;x}{(1 + sin\;x)^2}$

= cos x / (1 + sin x)²

e^x / 1 + sin x
y = e^x / 1 + sin x

$\frac{dy}{dx}\; = \frac{(1 + sin\;x)\frac{d}{dx}e^x\;-\;e^x\;\frac{d}{dx}(1 + sin\;x)}{(1 + sin\;x)^2}$

$=\frac{(1 + sin\;x).e^x\;-\;e^x.(0 + cos\;x)}{(1 + sin\;x)^2}$

$= \frac{e^x[1 + sin\;x - cos\;x]}{(1 + sin\;x)^2}$

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

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

$= \frac{1}{log\;5}\;.\;\frac{1}{\frac{log\;x}{log\;7}}\;.\;\frac{d}{dx}(\frac{log\;x}{log\;7})$

$= \frac{1}{log\;5}\;.\;\frac{1}{\frac{log\;x}{log\;7}}\;.\;\frac{1}{log\;7}\frac{d}{dx}(log\;x)$

$= \frac{1}{log\;5\;.\;.log\;x}\;.\;\frac{1}{x}$

= 1 / x log 5. log x

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

Solution:

y = log (3x² + 2x +1)

$= \frac{dy}{dx} = \frac{1}{3x^2 + 2x + 1}\;\;\frac{d}{dx}(3x^2 + 2x + 1)$

$= \frac{1}{3x^2+2x+1}\;\;(6x +2)$

$= \frac{2(3x +1)}{3x^2 + 2x +1}$

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

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

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

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

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

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

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

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

$f'(x)~=~\frac{cos\;x}{2\sqrt{sin\;x}}$

FAQs on Derivative of a Function

1. 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.

2. What is Dervivative of Sin x?

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

y'(x) = cos x

3. What is Derivative of Tan x?

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

y'(x) = sec² x

4. What is Derivative of Cos x?

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

y'(x) = -sin x

5. What is Derivative of e^x?

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

y'(x) = e^x

Suggest improvement

Algebra of Derivative of Functions

Share your thoughts in the comments

Derivative of a Function

What is Derivative of A Function

First Principle of Differentiation

Geometrical Interpretation of Derivative of a Function

How to Find Derivative of Function

Derivative of a Standard Functions

Derivative of Constant Function

Derivative of Exponential Function

Derivative of Trigonometric Functions

Derivative of Absolute Function

Derivative of Logarithmic Function

Derivative of Inverse Function Formula

Derivative of Implicit Function

Derivative of Composite Function

Algebra of Derivatives

Derivative Formulas

Examples on Derivative Rules

FAQs on Derivative of a Function

1. What is Derivative of a Function?

2. What is Dervivative of Sin x?

3. What is Derivative of Tan x?

4. What is Derivative of Cos x?

5. What is Derivative of ex?

Please Login to comment...

Similar Reads

What kind of Experience do you want to share?

5. What is Derivative of e^x?