Proof of Derivative of Sin Inverse x

The derivative of sin inverse x can be found by two methods:

• Using First Principle of Differentiation
• Using Implicit Differentiation

Derivative of Sin Inverse x by First Principle of Differentiation

The first principle of differentiation states that derivative of a function f(x) is defined as,

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

or

f'(x) = lim_h→0 [f (x + h) – f(x)]/ h

Putting f(x) = sin^-1x to find derivative of sin inverse x, we get,

f'(x) = lim_h→0 [sin^-1(x + h) – sin^-1(x)]/ h

Putting, A = sin^-1(x + h) and B = sin^-1(x), we get, h = sin A – sin B and limit changes to A →B,

⇒ f'(x) = Lim _A→B (A – B) / (sin A – sin B)

Using trigonometric relation, sin A – sin B = 2(sin(A-B)/2) (cos(A+B)/2) we get,

⇒ f'(x) = Lim _A→B (A – B)/2(sin(A-B)/2) × 1/cos(A+B)/2

Using limit relation, lim_x→0 (sin x)/x = 1, we get,

⇒ f'(x) = Lim _A→B 1/cos(A+B)/2

⇒ f'(x) = 1/cos B

⇒ f'(x) = 1/√(1-sin²B)

⇒ f'(x) = 1/√(1-x²)

Hence, the formula for derivative of sin inverse x has been derived using first principle of differentiation.

Derivative of Sin Inverse x Implicit Differentiation

Implicit differentiation is used for the functions represented as y = f(x), where it is complex to find derivative of f(x) and it is relatively easier to find the derivative of g(y). Hence, the function is represented as x = g(y). This method is useful for calculating derivative of inverse functions and logarithmic functions. The derivative of sin inverse x is derived using this method as follows.

Let, y = sin^-1x

Then, sin y = x

Differentiating on both sides of above equation, we get,

⇒ cos y dy = dx

⇒ dy/dx = 1/cos y

Now, cos y = √(1-sin²y) = √(1-x²)

⇒ dy/dx = 1/√(1-x²)

Thus, we have derived the derivative of sin inverse x using implicit differentiation.

Read More,

• Derivative of Inverse Trigonometric Functions
• Differentiation of Trigonometric Functions
• Derivative Formulas
• Derivative of Arcsin x

Derivative of sin inverse x is 1/√(1-x²). The derivative of any function gives the rate of change of the functional value with respect to the input variable. Sin inverse x is one of the inverse trigonometric functions. It is also represented as sin^-1x. There are inverse trigonometric functions corresponding to each trigonometric function. The derivative of a function also helps in finding the slope of the tangent to the curve represented by the function at any point.

In this article, we will learn about the derivative of sin inverse x, methods to find it including the first principle of differentiation and implicit differentiation, solved examples, and practice problems.