Hello

I have a function:

$\displaystyle y = \frac{e^x}{x}$

If I differentiate using the quotient rule, then:

let u = e^x and let v = x

$\displaystyle \frac{dy}{dx} = \frac{x.e^x - e^x.1}{x^2}$

$\displaystyle = \frac{e^x(x - 1)}{x^2}$

But could I not express as a product ie $\displaystyle y = e^x . x^-1$

And then use the product rule,

$\displaystyle let u = e^x and v = x^-1$

$\displaystyle \frac{dy}{dx} = x^-1 . e^x - e^x . x^-2$

$\displaystyle \frac{e^x}{x} - \frac{e^x}{x^2} = \frac{e^(2x)}{x^2} - \frac{e^x}{x^2}$

$\displaystyle = \frac{e^(2x) - e^x}{x^2}$

Which is a different result! Why so? Or have I done it wrong?