Limits, Continuity & Differentiability
Rules for Differentiation
Grade 12
Question:
<p>If <i>y</i> = sin <i>x</i> + <i>y</i>, then d<i>y</i>/d<i>x</i> is equal to</p>
<p>(a) cos <i>x</i>/(2<i>y</i> − 1)</p>
<p>(b) cos <i>x</i>/(1 − 2<i>y</i>)</p>
<p>(c) sin <i>x</i>/(1 − 2<i>y</i>)</p>
<p>(d) sin <i>x</i>/(2<i>y</i> − 1)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation to find d<i>y</i>/d<i>x</i> when both <i>x</i> and <i>y</i> appear in the equation.
<p><strong>Solution:</strong> Given <i>y</i> = sin <i>x</i> + <i>y</i>, rearrange to get <i>y</i> − <i>y</i> = sin <i>x</i>. Differentiate implicitly with respect to <i>x</i>: d<i>y</i>/d<i>x</i> − d<i>y</i>/d<i>x</i> = cos <i>x</i>. Solving for d<i>y</i>/d<i>x</i> yields cos <i>x</i>/(2<i>y</i> − 1).</p>
Correct Answer: A