Limits, Continuity & Differentiability
Rules for Differentiation
Grade 12
Question:
<p>If (<i>a</i> + 2<i>b</i> cos <i>x</i>)(<i>a</i> − 2<i>b</i> cos <i>y</i>) = <i>a</i><sup>2</sup> − <i>b</i><sup>2</sup>, where <i>a</i> > <i>b</i> > 0, then d<i>y</i>/d<i>x</i> at (π/4, π/4) is (JEE Main 2020)</p>
<p>(a) (<i>a</i> − <i>b</i>)/(<i>a</i> + <i>b</i>)</p>
<p>(b) (<i>a</i> + <i>b</i>)/(<i>a</i> − <i>b</i>)</p>
<p>(c) (2<i>a</i> + <i>b</i>)/(2<i>a</i> − <i>b</i>)</p>
<p>(d) (<i>a</i> − 2<i>b</i>)/(<i>a</i> + 2<i>b</i>)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation on the given constraint equation and evaluate the derivative at the specified point.
<p><strong>Solution:</strong> Expand the given equation and differentiate implicitly with respect to <i>x</i>. At the point (π/4, π/4), substitute cos(π/4) = 1/√2 and solve for d<i>y</i>/d<i>x</i> to get (<i>a</i> − <i>b</i>)/(<i>a</i> + <i>b</i>).</p>
Correct Answer: A