Test Index

NDA Math Differentiation

Directions (Q. Nos. 84 and 85): Read the following information and answer the two items that follow:

Consider the equation

x^y=e^{x–y}

[NDA II 2019]

Question : 20 of 64

Marks: +1, -0

What is

\frac{d^2y}{dx^2}

Solution:

\frac{dy}{dx}=\frac{\log x}{(1+\log x)^{2}}

from (ii)

\Rightarrow\frac{d^{2}y}{dx^{2}}=\frac{(1\log x)^{2}\cdot\frac{1}{x}-\log x\cdot 2(1+\log x)\cdot\left(0+\frac{1}{x}\right)}{(1+\log x)^{2}}

=\frac{(1+\log x)^{2}\cdot\frac{1}{x}-\frac{2}{x}\log x(1+\log x)}{(1+\log x)^{2}}

\therefore\left(\frac{d^{2}y}{dx^{2}}\right)_{x=1}=\frac{(1+\log 1)^{2}\cdot\frac{1}{1}-\frac{2}{1}\log 1(1+\log 1)}{(1+\log 1)^{2}}

=\frac{1-2\cdot0}{(1+0)^{2}}=\frac{1-0}{1}=1

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