William Boggs

Answered

2022-01-22

Implicit differentiation question
Given: $\frac{y}{x-y}={x}^{2}+1$

Answer & Explanation

Pansdorfp6

Expert

2022-01-22Added 27 answers

$\frac{y}{x-y}={x}^{2}+1$
You claim that ${y}^{\prime }=2x$
so that $y={x}^{2}+C$
This means $\frac{{x}^{2}+C}{x-{x}^{2}-C}={x}^{2}+1$
This is absqrt, since the quotient of two second degree polynomials can't be a second degree polynomial. In fact you get two non vanishing terms which are off.
I don't understand what your procedure is, also. I would proceed as follows:
$\frac{y}{x-y}={x}^{2}+1$
$\frac{d}{dx}\left(\frac{y}{x-y}\right)=\frac{d}{dx}\left({x}^{2}+1\right)$
$\frac{{y}^{\prime }\left(x-y\right)-\left(1-{y}^{\prime }\right)y}{{\left(x-y\right)}^{2}}=2x$
$\frac{{y}^{\prime }x-y}{{\left(x-y\right)}^{2}}=2x$
${y}^{\prime }x=2x{\left(x-y\right)}^{2}+y$
${y}^{\prime }=2{\left(x-y\right)}^{2}+\frac{y}{x}$

Barbara Meeker

Expert

2022-01-23Added 38 answers

An explicit approach:
Rewrite as $y=\left(x-y\right)\left({x}^{2}+1\right)$, and factor out y to get $y=\frac{{x}^{3}+x}{{x}^{2}+2}$, This is straightforward to differentiate, yielding $\frac{dy}{dx}=\frac{{x}^{4}+5{x}^{2}+2}{{\left({x}^{2}+2\right)}^{2}}$.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?