Thread: Calculus and Exponents

A question says to find the derivative of

$\displaystyle e^3/(sin(x)+1)$

The answer I got was $\displaystyle -e^3cosx/(sinx+1)^2$

My friend got
$\displaystyle (e^3sinx+e^3 - e^3cosx)/(sinx+1)^2$

The online calculators all agree with me, but he used the quotient rule (so did I), but I pulled e^3 out of it and used 1/sinx+1 to perform the quotient rule.

Which is the correct method?

I'd write it as $\displaystyle \displaystyle \begin{align*} y = e^3 \left[ \sin{(x)} + 1 \right] ^{-1} \end{align*}$ then let $\displaystyle \displaystyle \begin{align*} u = \sin{(x)} + 1 \implies y = e^3 \, u^{-1} \end{align*}$. Then $\displaystyle \displaystyle \begin{align*} \frac{du}{dx} = \cos{(x)} \end{align*}$ and $\displaystyle \displaystyle \begin{align*} \frac{dy}{du} = -e^3 \, u^{-2} = -e^3 \left[ \sin{(x)} + 1 \right] ^{-2} \end{align*}$. So $\displaystyle \displaystyle \begin{align*} \frac{dy}{dx} = -e^3 \cos{(x)} \left[ \sin{(x)} + 1 \right] ^{-2} = -\frac{ e^3 \cos{(x)} }{ \left[ \sin{(x)} + 1 \right] ^2} \end{align*}$. Your answer is correct.

Your friend failed to realize that $\displaystyle e^3$ is a constant, and so its derivative is zero.