JEECUP (UP Polytechnic) Implicit — practice questions

Explore 2 free Implicit questions in JEECUP (UP Polytechnic) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Implicit questions and exercises with answers

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If $x^2 + y^2 = 25$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $x^y = y^x$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →

Worked practice questions

If $x^2 + y^2 = 25$, then $\dfrac{dy}{dx}$ equals:

$x/y$
$-x/y$
$-y/x$
$y/x$

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Correct answer: B. $-x/y$

**Implicit differentiation:** $2x + 2y\,dy/dx = 0$ ⇒ $dy/dx = -x/y$. (Circle: slope of tangent.)

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If $x^y = y^x$, then $\dfrac{dy}{dx}$ equals:

$\dfrac{\ln y - \ln x}{x - y}$
$1$
$\dfrac{y(x \ln y - y)}{x(y \ln x - x)}$
$y/x$

Show answer and solution

Correct answer: C. $\dfrac{y(x \ln y - y)}{x(y \ln x - x)}$

Take $\ln$: $y \ln x = x \ln y$. **Differentiate:** $y'\ln x + y/x = \ln y + (x/y)y'$. Solve for $y'$: $y'(\ln x - x/y) = \ln y - y/x$ ⇒ $y' = (y(x\ln y - y))/(x(y\ln x - x))$.