Inverse Trig Derivative Examples
Example 1: Differentiate sin-1 (x)?
Solution:
Let, y = sinβ1 (x)
Taking sine on both sides of equation gives,
sin y = sin(sin-1x)
By the property of inverse trigonometry we know, sin(sin-1x) = x
sin y = x
Now differentiating both sides wrt to x,
d/dx{sin y} = d/dx{x}
{cos y}.dy/dx = 1
dy/dx = 1/ {cos y}
We can simplify it more by using the below observation:
sin2y + cos2y = 1
x2 + cos2y = 1 {As sin y = x}
cos2y = 1-x2
cos y = β(1 β x2)
Substituting the value, we get
dy/dx = 1/{cos y}
β dy/dx = 1/β(1 β x2)
Example 2: Differentiate cos-1 (x)?
Solution:
Let,
y = cosβ1 (x)
Taking cosine on both sides of equation gives,
cos y = cos(cos-1x)
By the property of inverse trigonometry we know, cos(cos-1x) = x
[Tex]cos (y) = x [/Tex]
Now differentiating both sides wrt to x,
d/dx{cos y} = d/dx{x}
{-sin y}.dy/dx = 1
dy/dx = -1/sin y
We can simplify it more by using the below observation:
sin2y + cos2y = 1
sin2y + x2 = 1 {As cos y = x}
sin2y = 1-x2
sin y = β(1 β x2)
Substituting the value, we get
dy/dx = -1/{sin y}
β dy/dx = -1/β(1 β x2)
Example 3: Differentiate tan-1 (x)?
Solution:
Let, y = tanβ1 (x)
Taking tan on both sides of equation gives,
tan y = tan(tan-1x)
By the property of inverse trigonometry we know, tan(tan-1x) = x
tan y = x
Now differentiating both sides wrt to x,
d/dx{sin y} = d/dx{x}
sec2(x).dy/dx= 1
dy/dx = 1/sec2x
We can simplify it more by using the below observation:
sec2y β tan2y = 1
sec2y β x2 = 1
sec2y = 1 + x2
Substituting the value, we get
dy/dx = 1/sec2y
dy/dx = 1/(1 + x2)
Example 4: y = cos-1 (-2x2). Find dy/dx at x = 1/2?
Solution:
Method 1 (Using implicit differentiation)
Given, y = cosβ1 (β2x2)
β cos y = β2x2
Differentiating both sides wrt x
d/dx{cos y} = d/dx{-2x2}
{-sin y}.dy/dx = -4x
dy/dx = 4x/sin y
Simplifying
sin2y + cos2y = 1
sin2y + (-2x2)2 = 1 {As cos y = -2x2}
sin2y + 4x4 = 1
sin2y = 1 β 4x4
sin y = β(1 β 4x4)
Putting the obtained value we get,
dy/dx = 4x/β{1 β 4x4}
β dy/dx = 4(1/2)/β{1 β 4(1/2)4}
β dy/dx = 2/β{1 β 1/4}
β dy/dx = 2/β{3/4}
β dy/dx = 4/β3
Method 2 (Using chain rule as we know the differentiation of cos inverse x)
Given, y = cosβ1 (β2x2)
Differentiating both sides wrt x
[Tex]\begin{aligned} \frac{dy}{dx} &=\frac{d}{dx} cos^{-1}(-2x^2) \\ &=\frac{-1}{\sqrt{1-(-2x^2)^2}}\ .\ (-4x) \\ &=\frac{4x}{\sqrt{1-4x^4}} \\ &=\frac{4(\frac{1}{2})}{\sqrt{1-4(\frac{1}{2})^4}} \\ &=\frac{2}{\sqrt{1-\frac{1}{4}}} \\ &=\frac{4}{\sqrt{3}} \end{aligned} [/Tex]
Example 5: Differentiate [Tex]\begin{aligned}sin^{-1}(\frac{1-x}{1+x}) \end{aligned} [/Tex]
Solutions:
Let,
[Tex]\begin{aligned} y = sin^{-1}(\frac{1-x}{1+x}) \end{aligned} [/Tex]
Differentiating both sides wrt x
[Tex]\begin{aligned} \frac{dy}{dx} &= \frac{d}{dx}sin^{-1}(\frac{1-x}{1+x}) \\ &= \frac{1}{\sqrt{1-(\frac{1-x}{1+x})^2}} \ . \frac{d}{dx}(\frac{1-x}{1+x}) \\ &= \frac{1+x}{\sqrt{(1+x)^2-({1-x})^2}} \ . \frac{-(1+x)-(1-x)}{(1+x)^2} \\ &= \frac{1}{\sqrt{(1+x)^2-({1-x})^2}} \ . \frac{-2}{(1+x)} \\ &= \frac{1}{\sqrt{4x}} \ . \frac{-2}{(1+x)} \\ &= \frac{-1}{\sqrt{x}(1+x)} \\ \end{aligned} [/Tex]
Derivative of Inverse Trig Functions
Derivative of Inverse Trig Function refers to the rate of change in Inverse Trigonometric Functions. We know that the derivative of a function is the rate of change in a function with respect to the independent variable. Before learning this, one should know the formulas of differentiation of Trigonometric Functions. To find the derivative of the Inverse Trigonometric Function, we will first equate the trigonometric function with another variable to find its inverse and then differentiate it using the implicit differentiation formula.
In this article, we will learn the Derivative of Inverse Trig Functions, Formulas of Differentiation of Inverse Trig Functions, and Solve some Examples based on it. But before heading forward, letβs brush up on the concept of inverse trigonometric functions and implicit differentiation.
Table of Content
- Inverse Trigonometric Functions
- What is Implicit Differentiation?
- What is Derivative of Inverse Trigonometric Functions?
- Proof of Derivative of Inverse Trig Functions
- Inverse Trig Derivative Formula
- Inverse Trig Derivative Examples
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