Question

If ${\left(f\left(x\right)\right)}^{g\left(y\right)}=e^{f\left(x\right)-g\left(y\right)}$ then $\frac{dy}{dx}$

• A. $\frac{f^{\prime }\left(x\right)\log f\left(x\right)}{g\left(y\right){(1+\log f(x\left)\right)}^2}$
• B. $\frac{f^{\prime }\left(x\right)\log f\left(x\right)}{g^1\left(y\right){(1+\log f(x\left)\right)}^2}$
• C. $\frac{f\left(x\right)\log f\left(x\right)}{g^{\prime }\left(y\right){(1-\log f(x\left)\right)}^2}$
• D. $2\frac{f^{\prime }\left(x\right)\log f\left(x\right)}{g\left(y\right){(1+\log f(x\left)\right)}^2}$

Answer 1

The correct option is B $\frac{f^{\prime }\left(x\right)\log f\left(x\right)}{g^1\left(y\right){(1+\log f(x\left)\right)}^2}$

Let $y=f\left(x\right)\Rightarrow \frac{dy}{dx}=f^{\prime }\left(x\right)$

Given:${\left(f\left(x\right)\right)}^{g\left(y\right)}=e^{f\left(x\right)-g\left(y\right)}$

Applying $\log$ to both sides,we get

$\log {\left(f\left(x\right)\right)}^{g\left(y\right)}=\log e^{f\left(x\right)-g\left(y\right)}$

$\Rightarrow g\left(y\right)\log \left(f\left(x\right)\right)=f\left(x\right)-g\left(y\right)$

$\Rightarrow g\left(y\right)\log \left(f\left(x\right)\right)+g\left(y\right)=f\left(x\right)$

$\Rightarrow g\left(y\right)(1+\log \left(f\left(x\right)\right))=f\left(x\right)$

$\Rightarrow g\left(y\right)=\frac{f\left(x\right)}{(1+\log \left(f\left(x\right)\right))}$

Differentiating both sides w.r.t $x$ we get

$\Rightarrow g^{\prime }\left(y\right)\frac{dy}{dx}=\frac{(1+\log \left(f\left(x\right)\right))f^{\prime }\left(x\right)-\frac{f\left(x\right)}{f\left(x\right)}{f}^{\prime }\left(x\right)}{{(1+\log \left(f\left(x\right)\right))}^2}$

$\Rightarrow g^{\prime }\left(y\right)\frac{dy}{dx}=\frac{(1+\log \left(f\left(x\right)\right))f^{\prime }\left(x\right)-f^{\prime }\left(x\right)}{{(1+\log \left(f\left(x\right)\right))}^2}$

$\Rightarrow g^{\prime }\left(y\right)\frac{dy}{dx}=\frac{f^{\prime }\left(x\right)+\log f\left(x\right)f^{\prime }\left(x\right)-f^{\prime }\left(x\right)}{{(1+\log f\left(x\right))}^2}$

$\Rightarrow g^{\prime }\left(y\right)\frac{dy}{dx}=\frac{\log f\left(x\right)f^{\prime }\left(x\right)}{{(1+\log f\left(x\right))}^2}$

$\therefore \frac{dy}{dx}=\frac{f^{\prime }\left(x\right)\log f\left(x\right)}{g^{\prime }\left(y\right){(1+\log f\left(x\right))}^2}$