Find (dy)/(dx) when 2x^2 + 3x^3y - 3y^2 = 44. (dy)/(dx) =...

Find \frac{dy}{dx} when $2x^2 + 3x^3y - 3y^2 = 44$. \frac{dy}{dx} = The slope of the tangent line to this curve at the point (2, 2) is The equation of the tangent line to this curve at this point is y =

Transcript

00:01 So what we'll have to do is some implicit differentiation.

00:05 But before i do that, i need to figure out when x equals 1, what y coordinate am i going to get? and i might get multiple y coordinates, but let's just solve for y.

00:18 Because if x is 1, 1 times y is y, and then 1 to the 4th power is still 1 times 32, it would give me 32.

00:27 So these two pieces would cancel.

00:29 If i divide 4 over, i get y cubed equals 8, and then the cube root of 8 would tell me that y equals 2.

00:38 May i'll just write it in black.

00:40 So that's the point that we're concerned with.

00:44 Let me write that again, is 1, 2.

00:47 So going back to this, i'm going to do implicit differentiation where i'm doing the power rule is 12 .y subtract 1 from the exponent, but then you have to write d, y, d, x, and then we have a product rule where the derivative of x is one, leave y alone.

01:02 Now you leave x alone, the derivative of y is d y d y d x, and then the derivative of y is d y d y d x.

01:11 And the derivative on the right side, i'm going to use a calculator that 32 times 4 is 128 x to the third power.

01:21 So i'm going to do two things at once.

01:23 One thing i'm going to do is subtract the 1y to the right side, and everything else i'm going to factor out the dydx.

01:32 So that's 12y squared plus x, each of these that have a dydx and a minus 1...

Question

Find \frac{dy}{dx} when $2x^2 + 3x^3y - 3y^2 = 44$. \frac{dy}{dx} = The slope of the tangent line to this curve at the point (2, 2) is The equation of the tangent line to this curve at this point is y =

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