Evaluate derivative with respect to x. (Help with Homework)

[surfelijo]

Can someone please help me with problem number 20:

Evaluate the following derivative with respect to x.

20. \(\displaystyle \dfrac{6y^3\, -\, x}{xy^2}\, =\, 5x\)

I have been working on it for about 40 minutes now. Thank you!.

 

[wjm11]

Can someone please help me with problem number 20:

Evaluate the following derivative with respect to x.

20. \(\displaystyle \dfrac{6y^3\, -\, x}{xy^2}\, =\, 5x\)

I have been working on it for about 40 minutes now. Thank you!.:smile:

Start by multiplying both sides by "xy^2", then use implicit differentiation (as you did in problem 19). Please show us what you've tried.

 

[Ishuda]

Can someone please help me with problem number 20:

Evaluate the following derivative with respect to x.

20. \(\displaystyle \dfrac{6y^3\, -\, x}{xy^2}\, =\, 5x\)

I have been working on it for about 40 minutes now. Thank you!.:smile:

This is just the quotient rule although there is a y' involved. Suppose we had

x = y³ / y²

and wanted to make life difficult for ourselves in finding y'. We have

\(\displaystyle x\, =\, 1\, =\, \left(\dfrac{y^3}{y^2}\right)' \,= \dfrac{ 3\, y^2 \,y'\, y^2 \,-\, y^3 \,2\, y\, y'}{y^4}\, =\, \dfrac{y^4\, y'\, (3\,-\,2)}{y^4}\)

or

y' = 1

which we would have got if we were smart and just cancelled the y² in numerator and denominator. Your problem is a little more complicated but it works in the same way.