Quiz #6 Math 2210-12 James Wibowo¶

if \(w(r)=\dfrac{3r+5}{\sqrt{r^2+12}}\) find \(w'(r)\)

Quotient Rule¶

\(\dfrac{w'(r)*g(r)-g'(r)*w(r)}{(w(r))^2}\)

\(g(r)=\sqrt{r^2+12}\) \(g(r)=(r^2+12)^{\frac{1}{2}}\) \(g'(r)=\frac{1}{2}(r^2+12)^{-\frac{1}{2}}\)

\(w(r)=3r+5\) \(w'(r)=3\)

\(\dfrac{\frac{d}{dr}(3r+5)*\sqrt{r^2+12}-(3r+5)*\frac{d}{dr}\left(\sqrt{r^2+12}\right)}{\left(\sqrt{r^2+12}\right)^2}\) \(\dfrac{3\sqrt{r^2+12}-(3r+5)*\frac{d}{dr}\left(\sqrt{r^2+12}\right)}{\left(\sqrt{r^2+12}\right)^2}\) ###### Power Rule \(n*w(r)^{n-1}*w'(r)\)

\(\frac{d}{dr}\left(\sqrt{r^2+12}\right)=\dfrac{1}{2}\left(r^2+12\right)^{\frac{1}{2}-1}*\frac{d}{dr}\left(r^2+12\right)\) \(\dfrac{3\sqrt{r^2+12}-\dfrac{(3r+5)*(2r+0)}{2\sqrt{r^2+12}}}{\left(\sqrt{r^2+12}\right)^2}\) \(\dfrac{3\sqrt{r^2+12}-\dfrac{r*\left(3r+5\right)}{\sqrt{r^2+12}}}{r^2+12}\) \(\dfrac{3}{\sqrt{r^2+12}}-\dfrac{r*(3r+5)}{\left(r^2+12\right)^\frac{3}{2}}\) \(\boxed{w'(r)=-\dfrac{5r-36}{\left(r^2+12\right)^\frac{3}{2}}}\)