I am a Calculus 2 student. I am doing implicit differentiation and I want to know the fastest way to simplify this and find y'. My online algebra calculator fails to ever solve problems the easy way. Hoping a math pro on here could show me "the easy way".
$$ \frac{x+y+y'}{xy}=e^{7x-y}(7-y') $$
$$ = $$
$$ \frac{1}{x}+\frac{y'}{y}=7e^{7x-y}-y'e^{7x-y} $$
$$ y'=? $$
Adding and subtracting messy fractions is never fun. Help friends :)
multiplying by $$xy$$ we get $$x+y+y'=xye^{7x-y}(7-y')$$ multiplying out: $$x+y'+y=7xye^{7x-y}-y'xye^{7x-y}$$ $$y'+y'xye^{7x-y}=7xye^{7x-y}-x-y$$ $$y'(1+xye^{7x-y})=7xye^{7x-y}-x-y$$ therefore $$y'=\frac{7xye^{7x-y}-x-y}{1+xye^{7x-y}}$$