Finding two part derivative/tangent line

Question

So I'm supposed to graph the given function at the tangent line at the given point, which I know how to do with an explicit/one part function but not an implicit/two part function.

The function is:
x^3+y^3=(9/2)xy and the given point is: (1,2)

I'm just having trouble finding the derivatives/knowing when to plug in the points.

Thank you!

Answer 1

Hi Morgan,

 

Use implicit differentiation to find y' (a formula for slopes of tangent lines to the given curve).

 

Something like:

 

3x² + 3y²y' = 4.5xy' + 4.5y  (product rule on right hand side)

 

Get terms with y' on LHS, other terms on the RHS, factor out the y' and then divide by the binomial after the y'.

 

y' = (9y - 6x²)/(6y² - 9x)  --this is with 4.5 returned to 9/2 and complex fraction simplified.

 

So, y'(1,2) = (18 - 6) / (24 - 9) = 12/15 = 4/5 --which is the slope of the tangent line to the curve at (1,2)

 

Now use point slope (or similar) to get the equation of the tangent line:

 

y-2 = 4/5 (x-1)

 

etc.

 

https://www.desmos.com/calculator/3kj81wvygl