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Derivative of an implicitly defined function at a point

This type of question, asked 3 times: 2021 Paper 18, 27 Jul 2022, 6 Apr 2023. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 18

    88
    If y=y(x)y=y(x) is an implicit function of xx such that log⁡e(x+y)=4xy\log_{e}(x+y) = 4xy, then d2ydx2\frac{d^{2}y}{dx^{2}} at x=0x= 0 is equal to

  2. Question 2

    27 Jul 2022

    176
    For the curve C:(x2+y2−3)+(x2−y2−1)5=0C:\left( x^{2}+y^{2}- 3 \right)+\left( x^{2}-y^{2}- 1 \right)^{5}= 0, the value of 3y′−y3y′′3y^{'}-y^{3}y^{''}, at the point (α,α),α>0(\alpha,\alpha),\alpha > 0, on CC, is equal to

  3. Question 3

    6 Apr 2023

    77
    If 2xy+3yx=202x^{y}+ 3y^{x}= 20, then dydx\frac{dy}{dx} at (2,2)(2,2) is equal to

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