KCET · Maths · Differentiation
If \( \tan ^{-1}\left(x^{2}+y^{2}\right)=\alpha \) then \( \frac{d y}{d x} \) is equal to
- A \( -\frac{x}{y} \)
- B \( x y \)
- C \( \frac{y}{x} \)
- D \(-x y \)
Answer & Solution
Correct Answer
(A) \( -\frac{x}{y} \)
Step-by-step Solution
Detailed explanation
Given that, \( \tan ^{-1}\left(x^{2}+y^{2}\right)=\alpha \)
\( \Rightarrow x^{2}+y^{2}=\tan \alpha \)
Differentiating with respect to \( x \), we get
\[
\begin{array}{l}
2 x+2 y \frac{d y}{d x}=0 \\
\Rightarrow \frac{d y}{d x}=\frac{-x}{y}
\end{array}
\]
\( \Rightarrow x^{2}+y^{2}=\tan \alpha \)
Differentiating with respect to \( x \), we get
\[
\begin{array}{l}
2 x+2 y \frac{d y}{d x}=0 \\
\Rightarrow \frac{d y}{d x}=\frac{-x}{y}
\end{array}
\]
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