derivative HELP w/ x^2 cos(y) + sin(2y) = xy

[yummymummy1713]

1) Find the derivative of:

. . .x^2 cos(y) + sin(2y) = xy

How on earth is this done? Implicit differentiation; yes, I know. But how exactly, is where I am confuzzled. Thanks!

Also:

2) Find the derivative of:

. . .y = [(x + 3)(x^2 + 1)^3 (x + 1)^2] / (x^2 + 10)^(1/2)

This gives me a headache. Where do you start?

 

[stapel]

1) Just apply the regular rules. When you get down to "x", leave it be; when you get down to "y", tack on a "dy/dx". For instance:

. . .d[x²cos(y)]/dx

. . . . .= (2x)cos(y) + (x²)(-sin(y))(dy/dx)

. . . . .= 2xcos(y) - x²sin(y)(dy/dx)

Apply the usual rules to each of the other terms. Once you've done with that, get all the "dy/dx" terms together on one side, factor out the "dy/dx", and then divide off everything else (to get "dy/dx" by itself). Then you're done.

2) Is your function as follows?

. . . . .\(\displaystyle \L y\, =\, \frac{(x\, +\, 3)(x^2\, +\, 1)^3 (x\, +\, 1)^2}{\sqrt{x^2\, +\, 10}}\)

If so, then just apply the Quotient Rule. (I don't know if you might find it easier to multiply things out on top first, and then differentiate. It'll probably be a mess, regardless.)

Eliz.

 

[yummymummy1713]

ok. for number 1 I see where you got

cos(y) * d/dx (x^2) + x^2 * d/dx (cos(y))

this gave you [ ( 2xcos(y) - x^2(sin(y))]

but what do I do to the other side? the = xy? and how do I simplify the above?

thanks.

 

[stapel]

what do I do to the other side? the = xy?

Apply the same rules (Product Rule, Chain Rule, etc) to the other terms.

how do I simplify the above?

Apply what you learned back in your algebra classes to work through the step-by-step instructions provided earlier.

Please reply showing what you have tried. Thank you.

Eliz.