Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.

Question

3x³=y 6, point (1, 2)

Answer 1

3x³y=6 (1,2) is on the graph of that function.

You could find the derivative explicitly by first solving for y: y = 2 x^-3, so y' = -6x^-4, when x = 1, y'=-6.

Using implicit differentiation requires the use of the product rule.

3x³y' + 9x²y = 0

y' = -9x²y/3x³= -3y/x

y'(1,2) = -6/1 = -6 (same result)

Visit this graph for confirmation and to see the equation of the tangent line at the point of tangency (1,2).

desmos.com/calculator/tvvm6k8sb4

Answer 2

If your question really means y⁶=3y³ then implicit differentiation gives you

6y⁵ (dy/dx) = 9x²

^{Then do a little algebra and substitute the given values.}

With your correction the problem becomes

3y ln 3x = ln 6

Then as Doug C. says you can either solve explicitly for y, differentiate and substitute OR

3y(3/3x)+ln 3x (3)(dy/dx)=0

Then do a little algebra and substitute.