Question:
(e) If \( y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that
\[
\frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a}.
\]
(e) If \( y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that
\[
\frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a}.
\]
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Implicit differentiation is essential when the dependent variable appears inside a trigonometric function.
Updated On: Mar 1, 2025
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Solution and Explanation
Differentiating both sides with respect to \( x \):
\[
\frac{dy}{dx} = \cos (a + y) - x \sin (a + y) \frac{dy}{dx}.
\]
Rearranging terms:
\[
\frac{dy}{dx} = \frac{\cos (a + y)}{1 + x \sin (a + y)}.
\]
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