Question

If $y=e^{ax}$, then show that $x\frac{dy}{dx}=y\log y$.

Answer 1

Given $y=e^{ax}$ .... (i)
or $\log y=ax$
Differentiating both sides of equation (i) w.r.t. $x$, we get
$\frac{1}{y}.\frac{dy}{dx}=a$
$\Rightarrow \frac{dy}{dx}=ay$
$\Rightarrow \frac{dy}{dx}=a{e}^{ax}$ ....$[\because y=e^{ax}]$
Multiplying both sides with $x$, we get
$x\frac{dy}{dx}=ax{e}^{ax}$
$\Rightarrow x\frac{dy}{dx}=y\log y$