Implicit Diff Example - # 3

[Jason76]

\(\displaystyle y \cos x = 2x^{2} + 5y^{2}\)

\(\displaystyle [\cos x][y'] + [y][-\sin x] = 4x + 10yy'\)

\(\displaystyle \cos xy' - \sin xy = 4x + 10yy'\)

\(\displaystyle -\sin xy = 4x + 10yy' - \cos xy'\)

\(\displaystyle -\sin xy = 4x + y'[10y - \cos x]\)

\(\displaystyle -\sin xy - 4x = y'[10y - \cos x]\)

\(\displaystyle \dfrac{-\sin xy - 4x}{10y - \cos x} = y'\)

\(\displaystyle y' = \dfrac{-\sin xy - 4x}{10y - \cos x}\)

 

[stapel]

Advice: Either use functional notation, or else re-order the factors. The expression "\(\displaystyle \cos xy'\)" is too easily misunderstood as "\(\displaystyle \cos(xy')\)". Better to use "\(\displaystyle y' \cos(x)\)" or at least "\(\displaystyle \cos(x) y'\)".

What is your question regarding this exercise?

 

[Jason76]

Got no clue on this one, need a few more pointers. Are u sure I didn't make some algebraic error? Keep typing in the answer, but it is wrong on the online homework.

 

[stapel]

Got no clue on this one, need a few more pointers.

You "got no clue" with respect to what? You've typed out quite a few lines of work and appear to have reached an answer. About what do you "need a few more pointers"?

Keep typing in the answer, but it is wrong on the online homework.

Unfortunately, we cannot troubleshoot your school's software. For all we know, the programming is expecting you to have simplified by reversing the two subtractions and cancelling off the two "minus" signs. That's a common sort of problem with something as brainless and unthinking as a computer program.

 

[srmichael]

\(\displaystyle y \cos x = 2x^{2} + 5y^{2}\)

\(\displaystyle [\cos x][y'] + [y][-\sin x] = 4x + 10yy'\)

\(\displaystyle \cos xy' - \sin xy = 4x + 10yy'\)

\(\displaystyle -\sin xy = 4x + 10yy' - \cos xy'\)

\(\displaystyle -\sin xy = 4x + y'[10y - \cos x]\)

\(\displaystyle -\sin xy - 4x = y'[10y - \cos x]\)

\(\displaystyle \dfrac{-\sin xy - 4x}{10y - \cos x} = y'\)

\(\displaystyle y' = \dfrac{-\sin xy - 4x}{10y - \cos x}\)

One of the difficulties with online problems, especially when it comes to derivatives, is that the answer you get and the answer the computer gets are the same but it's just that the simplification by either you, the computer or both don't jive.

The best you can do is simplify to what is "reasonable" and go with that.

For this one, your initial application of the implicit differentiation is correct, but it is the BA (Bad Algebra) after that and the errors that stapel pointed out that did you in.

\(\displaystyle y \cos x = 2x^{2} + 5y^{2}\)

\(\displaystyle [\cos x][y'] + [y][-\sin x] = 4x + 10yy'\)

\(\displaystyle y'\cos x - y\sin x = 4x + 10yy'\)

\(\displaystyle -y\sin x = 4x + 10yy' - y'\cos x\)

\(\displaystyle -y\sin x = 4x + y'[10y - \cos x]\)

\(\displaystyle -y\sin x - 4x = y'[10y - \cos x]\)

\(\displaystyle \dfrac{-y\sin x - 4x}{10y - \cos x} = y'\)

\(\displaystyle y' = \dfrac{-y\sin x - 4x}{10y - \cos x}\)

To me, this is a "reasonable" answer and I would end with that. The compute may go one step further and factor a -1 out from the numerator, but that should not be marked incorrect if you did not factor out a -1.