If `y=tan^(-1)x` , find `(d^2y)/(dx^2)` in terms of `y` alone.

Explanation

To find \(\frac{d^2y}{dx^2}\) in terms of \(y\) alone, given that \(y = \tan^{-1}(x)\), we will follow these steps:

### Step 1: Differentiate \(y\) with respect to \(x\)
We start with the function:
\[ y = \tan^{-1}(x) \]
The first derivative of \(y\) with respect to \(x\) is:
\[ \frac{dy}{dx} = \frac{1}{1 + x^2} \]

### Step 2: Differentiate \(\frac{dy}{dx}\) to find \(\frac{d^2y}{dx^2}\)
Next, we differentiate \(\frac{dy}{dx}\) to find the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{1}{1 + x^2}\right) \]
Using the quotient rule or recognizing it as \((1 + x^2)^{-1}\), we apply the chain rule:
\[ \frac{d^2y}{dx^2} = -\frac{1}{(1 + x^2)^2} \cdot \frac{d}{dx}(1 + x^2) = -\frac{1}{(1 + x^2)^2} \cdot 2x \]
Thus, we have:
\[ \frac{d^2y}{dx^2} = -\frac{2x}{(1 + x^2)^2} \]

### Step 3: Express \(x\) in terms of \(y\)
Since we need to express \(\frac{d^2y}{dx^2}\) in terms of \(y\), we use the relationship:
\[ y = \tan^{-1}(x) \implies x = \tan(y) \]

### Step 4: Substitute \(x\) in the second derivative
Now, we substitute \(x = \tan(y)\) into the second derivative:
\[ \frac{d^2y}{dx^2} = -\frac{2\tan(y)}{(1 + \tan^2(y))^2} \]
Using the identity \(1 + \tan^2(y) = \sec^2(y)\), we can rewrite the expression:
\[ \frac{d^2y}{dx^2} = -\frac{2\tan(y)}{\sec^4(y)} \]

### Step 5: Simplify the expression
Since \(\sec(y) = \frac{1}{\cos(y)}\), we have:
\[ \sec^4(y) = \frac{1}{\cos^4(y)} \]
Thus, we can simplify:
\[ \frac{d^2y}{dx^2} = -2\tan(y) \cos^4(y) \]
Also, recall that \(\tan(y) = \frac{\sin(y)}{\cos(y)}\):
\[ \frac{d^2y}{dx^2} = -2 \cdot \frac{\sin(y)}{\cos(y)} \cdot \cos^4(y) = -2 \sin(y) \cos^3(y) \]

### Final Result
The final expression for the second derivative in terms of \(y\) is:
\[ \frac{d^2y}{dx^2} = -2 \sin(y) \cos^3(y) \]