Derivative Of Sin Inverse X

Derivative of sin inverse x means the rate of change of sin inverse x with respect to x and it can be written as d(sin^-1x)/dx. Also, the process of finding the derivative of sin inverse x is known as the differentiation of sin inverse x. In this article, you will learn the formula for the derivative of sin inverse x, how to find the derivative of sin inverse x using the first principle of derivative along with solved examples.

Derivative of Sin Inverse x Formula

The formula for the derivative of sin^-1x is given by:

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How to Find the Derivative of Sin Inverse x?

The derivative of sin^-1x can be found using different techniques such as substitution, the first principle and so on. Let’s understand how to find the derivative of sin^-1x using the first principle of derivative.

Derivative of Sin Inverse x by First Principle

Let f(x) = sin^-1x

Using the First principle,

So,

Let us consider sin^-1(x + h) = A

⇒ sin A = x + h….(2)

And

sin^-1x = B

⇒ sin B = x….(3)

From (2) and (3),

h = sin B – sin A

Also, h → 0 means sin  A → sin B

Or

A → B for all A ∈ [-1, 1].

Now, substituting these equations in (1), we get;

Using the formula sin A – sin B = 2 cos(A + B)/2 sin(A – B)/2; we get;

This can be written as:

We know that, lim_x→0 x/sinx = 1,

So,

Thus,

From (3),

sin B = x

Using the identity sin²θ + cos²θ = 1,

cos B = √(1 – x²)

Therefore,

Derivative of Sin Inverse x Proof

We can find the derivative of sin inverse x using some differentiation formulas. The derivation of finding the derivative for sin^-1x is given below:

Let sin^-1(x) = y

sin y = x….(1)

Differentiating with respect to x on both sides,

d/dx (sin y) = dx/dx

cos y (dy/dx) = 1 

√(1 – x²) (dy/dx) = 1 [From (1), sin y = x and using sin²θ + cos²θ = 1, cos y = √(1 – x²)]

dy/dx = 1/√(1 – x²)

Therefoe, d(sin^-1x)/dx = 1/√(1 – x²)

Derivative of Sin Inverse x square

Let’s have a look at the process of evaluating the derivative of sin^-1(x2).

We know that, d(sin^-1x)/dx = 1/√(1 – x²)

Now, d(sin^-1x2)/dx = 1/√[1 – (x²)²] d(x²)/dx

= 2x/√(1 – x⁴)

Hence, the derivative of sin inverse x square is equal to 2x/√(1 – x⁴).

Solved Example

Question: Calculate the derivative of sin^-1(3x + 2).

Solution:

As we know,

d(sin^-1x)/dx = 1/√(1 – x²)

Now,

Practice Problems

1. Find the derivative of sin Inverse x with respect to cos inverse √(1 – x²).
2. Find the derivative of sin^-1(x/y).
3. Evaluate the derivative of sin^-1x with respect to cos^-1x.

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