Proof of Derivative of Arcsin x
The derivative of tan x can be proved using the following ways:
- By using Chain Rule
- By using the First Principle of Derivative
Derivative of Arcsin by Chain Rule
To prove derivative of Arcsin x by chain rule, we will use basic trigonometric and inverse trigonometric formula:
- sin2y + cos2y = 1
- sin (arcsin x) = x
Here is the proof of derivative of Arcsin x:
Let y = arcsinx
Taking sin on both sides
siny = sin(arcsinx)
By the definition of an inverse function, we have,
sin(arcsinx) = x
So the equation becomes siny = x …..(1)
Differentiating both sides with respect to x,
d/dx (siny) = d/dx (x)
cosy · d/dx(y) = 1 [ As d/dx(sin x) = cos x]
dy/dx = 1/cosy
Using one of the trigonometric identities
sin2 y+cos2 y = 1
∴ cos y = √1 – sin2y = √1–x2 [From (1) we have siny = x]
dy/dx = 1/√(1–x2)
Substituting y = arcsin x
d/dx (arcsinx) = arcsin′x = 1/√1 – x2
Also Check, Chain Rule
Derivative of Arcsin by First Principle
To prove derivative of arcsin x using First Principle of Derivative, we will use basic limits and trigonometric formulas which are listed below:
- sin2 y+cos2 y = 1
- limx→0 x/sinx = 1
- sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2]
We can prove the derivative of arcsin by First Principle using the following steps:
Let f(x) = arcsinx
By First principle we have
[Tex] \frac{d f( x)}{dx} =\displaystyle \lim_{h \to 0} \frac{f (x + h)- f(x)}{h}[/Tex]
put f(x) = arcsinx, we get
[Tex]\frac{d}{dx}(arcsin x) =\displaystyle \lim_{h \to 0} \frac{arcsin (x + h)- arcsin x}{h}[/Tex]….(1)
Assume that arcsin (x + h) = A and arcsin x = B
So we have,
sin A = x+h …..(2)
sin B = x …….(3)
Subtract (3) from (2), we have
sin A – sinB = (x+h) – x
sinA – sinB = h
If h → 0, (sin A – sin B) → 0
sin A → sin B or A → B
Substitute these values in eq(1)
[Tex]\frac{d}{dx}(arcsin x) =\displaystyle \lim_{A \to B} \frac{A- B}{Sin A- Sin B}[/Tex]
Using sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2], we get
[Tex]\frac{d}{dx}(arcsin x) =\displaystyle \lim_{A \to B} \frac{A- B}{2Cos \frac{A+B}{2}- 2 Sin \frac{A-B}{2}}[/Tex]
which can be written as:
[Tex]\frac{d}{dx}(arcsin x) =\displaystyle \lim_{A \to B} \frac{\frac{A- B}{2}}{Sin \frac{A-B}{2}}\times \frac{1}{Cos \frac{A+B}{2}}[/Tex]
Now, we know limx→0 x/sinx = 1, therefore the above equation changes to
[Tex]\frac{d}{dx}(arcsin x) ={1}\times \frac{1}{Cos \frac{B+B}{2}}[/Tex]
[Tex]\frac{d}{dx}(arcsin x) =\frac{1}{Cos {B}} [/Tex]
Using one of the trigonometric identities
sin2 y+cos2 y = 1
∴ cos B = √1 – sin2B = √1–x2 [Sin B = x from (3)]
f′(x) = dy/dx = 1 / √(1–x2)
Also, Check
Derivative of Arcsin
Derivative of Arcsin x is d/dx(arcsin x) = 1/√1-x². It is denoted by d/dx(arcsin x) or d/dx(sin-1x). Derivative of Arcsin refers to the process of finding the rate of change in Arcsin x function with respect to the independent variable. Derivative of Arcsin x is also known as differentiation of Arcsin.
In this article, we will learn about the derivative of Arcsin and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule method.
Table of Content
- What is Derivative in Math?
- What is Derivative of Arcsin x?
- Proof of Derivative of Arcsin x
- Solved Examples on Derivative of Arcsin x
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