Question

Find the indicated derivative using implicit differentiation. HINT [See Example 1.]

e^x/y^2 = 8 + e^y; dy/dx

Answer 1

Rewrite equation as:
\[e^x=8y^2+y^2e^y\]
Take d/dx of each side of this equation using product rule for the second term on the right hand side:
\[e^x=16y \frac{ dy }{ dx }+2ye^y \frac{ dy }{ dx }+y^2e^y \frac{ dy }{ dx }\]
Factor out the dy/dx:
\[e^x=(16y+2ye^y+y^2e^y) \frac{ dy }{ dx }\]
Divide each side by the expression in the brackets on the right to get the answer:
\[\frac{ dy }{ dx }=\frac{ e^x }{ (16y+2ye^y+y^2e^y) }\]