Question

Find the slope of the tangent line equation of the curve
x^2 lny+xy^3= 5
at the point (5,1) on the curve by using implicit differentiation

Answer 1

\[\begin{align*}
x^2\ln y+xy^3&=5\\\\
\frac{d}{dx}\left[x^2\ln y+xy^3\right]&=\frac{d}{dx}5\\\\
\frac{d}{dx}\left[x^2\ln y\right]+\frac{d}{dx}\left[xy^3\right]&=0\\\\
x^2\frac{d}{dx}\left[\ln y\right]+\ln y\frac{d}{dx}\left[x^2\right]+y^3\frac{d}{dx}\left[xy^3\right]+x\frac{d}{dx}\left[y^3\right]&=0\\\\
\cdots&=0
\end{align*}\]
Any problems so far?

Answer 2

\[\frac{d}{dx}(xy^3)=x \frac{d}{dx}(y^3)+y^3 \frac{d}{dx}(x)\]