Doug C. answered • 07/18/17

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Hi Don,

Need to find the slope of the tangent line at (2/3, 4/3). Since the original equation cannot be easily solved for y, use implicit differentiation. The understanding is that y is some function of x, whenever we take the derivative of y it is with respect to x, i.e. dy/dx. In the following y' is used instead of dy/dx.

On the left hand side use the product rule where the "first function is 3x" and the second is y, at least that is one way to think about it.

3xy' + 3y = 3x

^{2}+ 3y^{2}y'Notice at this point we can divide all terms by 3 as part of the simplification. Transposing terms containing y' to left side and others to right side we have:

xy' - y

^{2}y' = x^{2}- yFactor out the y' from both terms on the left. Divide the right side by the coefficient of y'. Now use (2/3, 4/3) in the formula for y' to get the slope of the tangent line at that point (4/5). Now use point-slope to write the equation.

Here is a picture:

https://www.desmos.com/calculator/uhxhft0zvl

Don J.

For the equation I got

07/18/17