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

Explanation

To solve the problem of finding the second derivative of \( y = \tan^{-1} x \) in terms of \( y \) alone, we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)

Given:
\[ y = \tan^{-1} x \]

The first derivative is:
\[ \frac{dy}{dx} = \frac{1}{1 + x^2} \]

### Step 2: Differentiate again to find the second derivative

Now, we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{1}{1 + x^2} \right) \]

Using the quotient rule or the chain rule, we get:
\[ \frac{d^2y}{dx^2} = -\frac{2x}{(1 + x^2)^2} \]

### Step 3: Express \( x \) in terms of \( y \)

From the original equation \( y = \tan^{-1} x \), we can express \( x \) as:
\[ x = \tan y \]

### Step 4: Substitute \( x \) into the second derivative

Now, substitute \( x = \tan y \) into the second derivative:
\[ \frac{d^2y}{dx^2} = -\frac{2\tan y}{(1 + \tan^2 y)^2} \]

### Step 5: Simplify using trigonometric identities

Using the identity \( 1 + \tan^2 y = \sec^2 y \), we can rewrite the expression:
\[ \frac{d^2y}{dx^2} = -\frac{2\tan y}{(\sec^2 y)^2} \]

This simplifies to:
\[ \frac{d^2y}{dx^2} = -\frac{2\tan y}{\sec^4 y} \]

### Step 6: Further simplification

Since \( \tan y = \frac{\sin y}{\cos y} \) and \( \sec y = \frac{1}{\cos y} \), we can express this as:
\[ \frac{d^2y}{dx^2} = -\frac{2 \sin y}{\cos y} \cdot \cos^4 y \]

This simplifies to:
\[ \frac{d^2y}{dx^2} = -2 \sin y \cos^3 y \]

### Step 7: Use the double angle formula

Using the double angle formula \( 2 \sin y \cos y = \sin 2y \):
\[ \frac{d^2y}{dx^2} = -\sin 2y \cos^2 y \]

### Final Answer

Thus, the second derivative of \( y \) with respect to \( x \) in terms of \( y \) alone is:
\[ \frac{d^2y}{dx^2} = -\sin 2y \cos^2 y \]

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