Given x= and y=, find / x.

$\frac{x\ln\psi}{\ln\phi\ln\psi-1}$

$\frac{x\ln\phi}{\ln\phi\ln\psi-1}$

$\frac{\ln\phi}{\ln\phi\ln\psi-1}$

GATE 2019 CE (CE1) – Question 40

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 2 marks · Multiple choice

Given $x=\psi\ln\phi$ and $y=\phi\ln\psi$, find $\partial\psi/\partial x$.

$\frac{x\ln\psi}{\ln\phi\ln\psi-1}$
$\frac{x\ln\phi}{\ln\phi\ln\psi-1}$
$\frac{\ln\phi}{\ln\phi\ln\psi-1}$
$\frac{\ln\psi}{\ln\phi\ln\psi-1}$

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Show answer and explanation

Correct answer: (D) $\frac{\ln\psi}{\ln\phi\ln\psi-1}$

Explanation

Hold y fixed. From $y=\phi\ln\psi$, $0=\ln\psi\,d\phi+(\phi/\psi)d\psi$, so $d\phi/d\psi=-\phi/(\psi\ln\psi)$. Differentiate $x=\psi\ln\phi$: $$\frac{dx}{d\psi}=\ln\phi+\frac\psi\phi\frac{d\phi}{d\psi}=\ln\phi-\frac1{\ln\psi}.$$ Invert: $\partial\psi/\partial x=\ln\psi/(\ln\phi\ln\psi-1)$, **D**, with y held fixed.