Consider two function \frac{𝜕 \Psi}{𝜕x}:: x=\Psi \ln ϕ and y=ϕ \ln \Psi. Which one of the following is the correct expression for ?

Correct Answer is : lnΨlnϕlnΨ−1

\frac{x \ln ϕ}{\ln ϕ \ln \Psi-1}

\frac{\ln ϕ}{\ln ϕ \ln \Psi-1}

\frac{\ln \Psi}{\ln ϕ\ln \Psi-1}

\frac{x \ln \psi}{\ln ϕ \ln \psi-1}

GATE Civil Engineering (CE) 2019 Shift 1 Solved Paper

Question : 63 of 65

Marks: +1, -0

Consider two function

\;{𝜕 Ψ}/{𝜕x}:: x=Ψ \ln ϕ

and

y=ϕ \ln Ψ

. Which one of the following is the correct expression for ?

Solution:

x=Ψ \ln ϕ ⇒ Ψ \;{x}/{\ln ϕ}

.....(i)

y=ϕ\ln Ψ ⇒ ϕ \;{y}/{\ln Ψ}

Putting value of

ϕ

in (i)

Ψ=\;{ {x}}/{\ln (\;{ {y}}/{\In Ψ})}=\;{ {x}}/{\ln γ-\ln (\ln Ψ)}

...(ii)
Assuming y constant and differentiating

Ψ

w.r.t. x

\;{∂ Ψ}/{∂ x}=\;{(\ln y-\ln (\ln Ψ s)) ⋅ 1-x(0-{1}/{\ln Ψ} ⋅ {1}/{Ψ} ⋅ {∂ Ψ}/{∂ x})}/{(\ln y-\ln (\ln Ψ))^{2}}

......(iii)

Putting value of

(\ln y-\ln (\ln Ψ))=\;{x}/{Ψ}

from (ii) in equation (iii)

\;{𝜕 Ψ}/{∂ x}=\;{\;{x}/{Ψ}+x × {1}/{Ψ \ln Ψ} ⋅ {∂ Ψ}/{∂ x}}/{(\;{x}/{Ψ})^{2}}

⇒ \;\; \;{∂ Ψ}/{∂ x}=\;{1+{1}/{Ψ \ln Ψ} {∂ Ψ}/{∂ x}}/{(\;{x}/{Ψ})}

⇒ \;\; \;{x}/{Ψ} \;{∂ Ψ}/{∂ x}=1+\;{1}/{\ln Ψ} \;{∂ Ψ}/{∂ x}

⇒ \;\; \;{∂ Ψ}/{∂ x}(\;{x}/{Ψ}-\;{1}/{\ln Ψ})=1

⇒ \;\; \;{∂ Ψ}/{∂ x}=\;{1}/{{X}/{Ψ}-\;{1}/{\ln Ψ}}=\;{Ψ \ln Ψ}/{X \ln Ψ-Ψ}

⇒ \;\; \;{∂ Ψ}/{∂ x}=\;{Ψ \ln Ψ}/{Ψ \ln ϕ \ln Ψ-Ψ}

(

replacing

{x}

Ψ

ϕ)

⇒\;{∂ Ψ}/{∂ {x}}=\;{\ln Ψ}/{\ln ϕ \ln Ψ-1}

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