i hate derivatives....

[jmsic]

I am so confused... im used to always writting f prime as my form for derivatives but my teacher put dy/dx on a practice test and i was like.... uhhh...
so the problems are:
Given x[sup:38waewiw]2[/sup:38waewiw] + xy = 5 find d[sup:38waewiw]2[/sup:38waewiw]y/dx[sup:38waewiw]2[/sup:38waewiw] in terms of x and y.

and

Find dy/dx given that y[sup:38waewiw]3[/sup:38waewiw] + y[sup:38waewiw]2[/sup:38waewiw] - 5y - x[sup:38waewiw]2[/sup:38waewiw] = -4

and

At time t = 0, a diver jumps from a diving board that is 32 feet above the water. The position of the diver is given by s(t)= -16t[sup:38waewiw]2[/sup:38waewiw]+16t+32 where s is measured in feet and t is measured in seconds. Find the diver’s velocity when he lands in the water

 

[galactus]

This is an exercise in implicit differentiation, not a differential equation.

dy/dx or y' is just a matter of notation. I will use y' then.

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

Remember the product rule on xy:

\(\displaystyle 2x+\underbrace{xy'+y}_{\text{product rule on xy}}=0\)

Solve for y':

\(\displaystyle y'=\frac{-(y+2x)}{x}\)

Now, to find y'', differentiate that again (quotient rule) and sub in the above for y'.