Derivative of Arcsin

Derivative of Arcsin x is 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.

What is Derivative in Math?

Derivative of a function is the rate of change of the function with respect to any independent variable. The derivative of a function f(x) is denoted as f'(x) or (d /dx)[f(x)]. The differentiation of a trigonometric function is called a derivative of the trigonometric function or trig derivatives. The derivative of a function f(x) is defined as:

f'(x₀) = lim_h→0 [f(x₀ + h) – f(x₀)] / h

What is Derivative of Arcsin x?

Among the trig derivatives, the derivative of the Arcsin x is one of the derivatives. Derivative of arcsin function represents the rate at which the arcsin curve is changing at a given point. It is denoted by d/dx(arcsin x) or d/dx(sin^-1x). Arcsinx is also known as inverse sin x.

Derivative of the Arcsin x is 1/√1-x²

Derivative of Arcsin x Formula

The formula for the derivative of Arcsin x is given by:

(d/dx) [Arcsin x] = 1/√1-x²

OR

(Arcsin x)’ = 1/√1-x²

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 trignometric 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

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

• lim_x→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

put f(x) = arcsinx, we get

….(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)

Using sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2], we get

which can be written as:

Now, we know lim_x→0 x/sinx = 1, therefore the above equation changes to

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)

Solved Examples on Derivative of Arcsin x

Example 1: Find the derivative of y = arcsin (3x).

Solution:

Let f(x) = arcsin (3x).

We know that d/dx (arcsin x) = 1/√1 – x².

By chain rule,

d/dx(arcsin(3x)) = 1/√(1 – (3x)² · d/dx (3x)

= 1/ √(1 -9x²) · (3)

= 3/√(1 -9x²)

Hence, the derivative of y = arcsin (3x) is 3/√(1 -9x²).

Example 2: Find the derivative of y = arcsin (1/2x).

Solution:

Let f(x) = arcsin (1/2x).

We know that d/dx (arcsin x) = 1/√1 – x².

By chain rule,

d/dx(arcsin(1/2x)) = 1/√(1 – (1/2x)² · d/dx (1/2x)

= 1/ √(1 -(1/4x²) )· (-1/2x²)

= 1/√(4x² – 1)/4x² · (-1/2x²)

= -1/x√4x² – 1

Hence, the derivative of y = arcsin (1/x) is -1/x√4x² – 1.

Example 3: Find the derivative of y = x arcsin x.

Solution:

We have y = x arcsin x.

d/dx(arcsin(1/x)) = x · d/dx (arcsin x) + arcsin x · d/dx (x)

= x [1/√1-x²] + arcsin x (1)

= x/√1-x² + arcsin x
Hence, the derivative of y = arcsin (1/x) is x/√1-x² + arcsin x

Practice Questions on Derivative of Sin x

Q1: Find the derivative of arcsin(5x).

Q2: Find the derivative of x³ arcsin(x).

Q3: Evaluate: d/dx [ arcsin(x) / x² + 1 ]

Q4: Evaluate the derivative of arcsin(x) -tan(x)

Derivative of Arcsin FAQs

What is Derivative of Arcsin?

Derivative of the Arcsin x is 1/√1-x²

What is Derivative in Math?

In mathematics, the derivativeis the measures how a function changes as its input (independent variable) changes. The derivative of a function f(x) is denoted as f'(x) or (d /dx)[f(x)].

What is Derivative of arcsin(1/x)?

The derivative of the arcsin(1/x) is (-1) / (x√x² – 1).

What is Derivative?

Derivative of function is defined as the rate of change of the function with respect to an independent variable.

What is Derivative of sinx?

Derivative of sinx is cosx.