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.
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
- Types of Derivatives
- First Order Derivative
- Second Order Derivative
- nth Order Derivative
- Derivatives Formula
- Rules of Derivatives
- Derivative of Composite Function
- Chain Rule of Derivatives
- Derivative of Implicit Function
- Parametric Derivatives
- Higher Order Derivatives
- Partial Derivative
- Logarithmic Differentiation
- Applications of Derivatives
- Derivatives Examples
- Sample Problems on Derivatives
- Practice Problems on 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.
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