Q.Differentiate w.r.t. $x$: $\sin^{-1}\left(\dfrac{1}{\sqrt{x + 1}}\right)$.

Concept: Chain Rule – differentiate the outer inverse sine function, then multiply by the derivative of the inner function.

Let $y = \sin^{-1}\left(\dfrac{1}{\sqrt{x + 1}}\right)$.

Step 1: Outer derivative: $\frac{d}{du} \sin^{-1} u = \frac{1}{\sqrt{1 - u^2}}$, where $u = \frac{1}{\sqrt{x+1}}$.

Step 2: Compute $u^2 = \frac{1}{x+1}$, so $1 - u^2 = \frac{x}{x+1}$. Thus $\sqrt{1-u^2} = \sqrt{\frac{x}{x+1}}$.

Step 3: Inner derivative: $\frac{du}{dx} = -\frac{1}{2}(x+1)^{-3/2} = -\frac{1}{2 (x+1)^{3/2}}$. …

We need to differentiate $y = \sin^{-1}\left(\frac{1}{\sqrt{x+1}}\right)$ with respect to $x$. The key here is the Chain Rule: when you have a function of a function, you differentiate the outer function first, then multiply by the derivative of the inner function.

The outer function is $\sin^{-1} u$, whose derivative is $\frac{1}{\sqrt{1-u^2}}$. The inner function is $u = \frac{1}{\sqrt{x+1}}$, which itself is a composition — a reciprocal of a square root. So we’ll need the Chain Rule twice.

Let’s work through it step by step.

1. Set up the Chain Rule. If $y = \sin^{-1} u$ and $u = \frac{1}{\sqrt{x+1}}$, then

We know $\frac{dy}{du} = \frac{1}{\sqrt{1-u^2}}$.

2. Find $\frac{du}{dx}$. Write $u = (x+1)^{-1/2}$. Differentiate using the power rule:

3. Substitute $u$ into $\frac{dy}{du}$. We have $u = \frac{1}{\sqrt{x+1}}$, so

Then

Therefore

4. Multiply the derivatives.

Simplify:

So …

Method: Chain Rule for an Inverse Sine of a Composite Argument

This method applies whenever you must differentiate $\sin^{-1}(u)$ where $u$ is itself an algebraic function of $x$ — the general skill is separating the "outer" inverse-trig layer from the "inner" algebraic layer and simplifying before you multiply.

Steps

Step 1: Identify the outer and inner functions

Write the given expression as $y = \sin^{-1}(u)$ where $u$ is everything inside the inverse sine. Here $u = \dfrac{1}{\sqrt{x+1}} = (x+1)^{-1/2}$.

Step 2: Recall the outer derivative

Step 3: Differentiate the inner function

Rewrite $u$ as a power of $(x+1)$ and use the power rule:

Step 4: Simplify $1-u^2$ algebraically before combining …

Common Mistakes

Mistake 1: Leaving $\sqrt{1-u^2}$ as a nested radical instead of simplifying first

Why it's wrong: without reducing $1-u^2=\dfrac{x}{x+1}$ to a single fraction before taking the square root, the algebra becomes unmanageable and students often make a sign or power error while multiplying by $\dfrac{du}{dx}$. Correct approach: always reduce $1-u^2$ to a single fraction first, then split the square root of a quotient as $\sqrt{a}/\sqrt{b}$.

Mistake 2: Ignoring the domain restriction implied by $\sin^{-1}$ …