If 2 x^y+3 y^x=20, then [(d y)/(d x)] at (2,2) is equal to :

Question

If $2 x^{y}+3 y^{x}=20$, then $$\frac{d y}{d x}$$ at $(2,2)$ is equal to :

Accepted Answer

Given, $2 x^y+3 y^x=20$ ..........(i)

Let $u=x^y$

On taking log both sides, we get

$\log u=y \log x$

On differentiating both sides with respect to $x$, we get

$$
\[\begin{gathered} \frac{1}{u} \frac{d u}{d x}  =y \frac{1}{x}+\log x \frac{d y}{d x} \ \Rightarrow  \frac{d u}{d x}  =u\left(\frac{y}{x}+\log x \frac{d y}{d x}\right) \ \Rightarrow  \frac{d u}{d x}  =x^y\left(\frac{y}{x}+\log x \frac{d y}{d x}\right) .........(ii) \end{gathered}$$
\]

Also, let $v=y^x$

On taking log both sides, we get

$\log v=x \log y$

On differentiating both sides, we get

$$
\[\begin{gathered} \frac{1}{v} \frac{d v}{d x}  =x \frac{1}{y} \frac{d y}{d x}+\log y \cdot 1 \ \Rightarrow \frac{d v}{d x}  =v\left(\frac{x}{y} \frac{d y}{d x}+\log y\right) \ \Rightarrow \frac{d v}{d x}  =y^x\left(\frac{x}{y} \frac{d y}{d x}+\log y\right) ..........(iii) \end{gathered}$$
\]

Now, from Equation (i), $2 u+3 v=20$

$$
\begin{aligned}
& \Rightarrow 2 \frac{d u}{d x}+3 \frac{d v}{d x}=0 \\
& \Rightarrow 2 x^y\left(\frac{y}{x}+\log x \frac{d y}{d x}\right)+3 y^x\left(\frac{x}{y} \frac{d y}{d x}+\log y\right)=0 \text { [Using Eqs. (ii) and (iii)] }
\end{aligned}
$$

On putting $x=2$ and $y=2$, we get

$$
\begin{aligned}
& 2(4)\left(1+\log 2 \frac{d y}{d x}\right)+3(4)\left(\frac{d y}{d x}+\log 2\right)=0 \\
& \Rightarrow \frac{d y}{d x}(8 \log 2+12)+(8+12 \log 2)=0 \\
& \Rightarrow \frac{d y}{d x}=-\left(\frac{2+3 \log 2}{3+2 \log 2}\right) \\
& \Rightarrow \frac{d y}{d x}=-\left(\frac{2+\log 8}{3+\log 4}\right)
\end{aligned}
$$

Solutions · 1