If $z=tan^{-1}\frac{y}{x}$, then $dz=$

Question

Accepted Answer

To determine the differential $ dz $ when $ z = \tan^{-1}\left(\frac{y}{x}\right) $, we need to apply implicit differentiation and the chain rule. Let's go through the steps to derive the correct expression for $ dz $.

### Derivation of $ dz $

1. **Expression for $ z $:**
   $$
   z = \tan^{-1}\left(\frac{y}{x}\right)
   $$

2. **Differentiating both sides with respect to $ x $:**
   Using the chain rule, the derivative of $ \tan^{-1}(u) $ with respect to $ u $ is $ \frac{1}{1+u^2} $. Here, $ u = \frac{y}{x} $.

3. **Apply the chain rule:**
   $$
   dz = \frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{d}{dx}\left(\frac{y}{x}\right)
   $$

4. **Differentiate $ \frac{y}{x} $:**
   Using the quotient rule, where $ u = y $ and $ v = x $, we have:
   $$
   \frac{d}{dx}\left(\frac{y}{x}\right) = \frac{x \cdot dy - y \cdot dx}{x^2}
   $$

5. **Substitute back into the expression for $ dz $:**
   $$
   dz = \frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{x \cdot dy - y \cdot dx}{x^2}
   $$

6. **Simplify the expression:**
   $$
   dz = \frac{x \cdot dy - y \cdot dx}{x^2 + y^2}
   $$

This matches Option 1: $\frac{x~dy-y~dx}{x^{2}+y^{2}}$.

### Explanation of Incorrect Options

- **Option 2: $\frac{x~dy+y~dx}{x^{2}+y^{2}}$**
  - This option incorrectly adds $ y \cdot dx $ instead of subtracting it. The correct application of the quotient rule results in a subtraction, as shown in the derivation.

- **Option 3: $\frac{x~dx+y~dy}{x^{2}+y^{2}}$**
  - This option incorrectly uses $ x \cdot dx + y \cdot dy $, which does not follow from the differentiation of $ \frac{y}{x} $. The terms should involve $ x \cdot dy $ and $ y \cdot dx $ due to the quotient rule.

- **Option 4: $\frac{x~dx-y~dy}{x^{2}+y^{2}}$**
  - This option incorrectly uses $ x \cdot dx - y \cdot dy $, which does not align with the differentiation of $ \frac{y}{x} $. The correct terms involve $ x \cdot dy $ and $ y \cdot dx $.

In conclusion, Option 1 is the correct answer because it accurately reflects the result of differentiating $ z = \tan^{-1}\left(\frac{y}{x}\right) $ using the chain rule and quotient rule. The other options do not correctly apply these rules.