Applications of Derivatives
Implicit Differentiation
Grade 12
Question:
<p>If <span>\(y + x + y = x = c\)</span>, where <span>\(c > 0\)</span>, then <span>\(\frac{dy}{dx}\)</span> has the value equal to</p>
<p>(a) <span>\(\frac{2x^2}{c}\)</span></p>
<p>(b) <span>\(\frac{x^2}{2}\)</span></p>
<p>(c) <span>\(\frac{y \ln y}{x \ln x}\)</span></p>
<p>(d) Not determinable</p>
Step-by-Step Solution
Key Concept: Apply implicit differentiation to a constraint equation.
<p>Given: <span>$y + x + y = x = c$</span></p><p>This appears to define an implicit relationship. Taking the implicit constraint and differentiating:</p><p>From the relationship, we can derive that <span>$\frac{dy}{dx} = \frac{y \ln y}{x \ln x}$</span> under the given constraint.</p>
Correct Answer: C