Use implicit differentiation to find the slope of the tangent line to the curve (4x^2) + (xy) - (y^3) = 72​ at the point ( 2 , − 4 )​. If the slope is undefined, enter 'undefined'

Answer 1

4x² + xy - y³ = 72

The derivative of 4x² with respect to x is 8x (power rule)

The derivative of xy with respect to x is found by using the product rule (u•v)' = u'v + uv') so it is x'y + xy' = y + xy'

The derivative of -y³ with respect to x is -3y²•y' (using the power rule and chain rule)

The derivative of 72 with respect to x is 0 (zero)

Putting these together we get:

8x + y + xy' - 3y²•y' = 0

8x + y = 3y²•y' - xy'

8x + y = y'(3y² - x)

(8x + y)/(3y² - x) = y'

For the point (2, -4) then:

y' = (8•2 - 4)/(3•16 - 2) = 12/46 = 6/23 ←this is the slope of the tangent line

Answer 2

8x + y + xy' - 3y²y' = 0; at point (2, - 4) we have: 16 - 4 + 2y' - 48y' = 0; 46y' = 12; y' = 6/23;

Tangent line: y = - 4 + 6/23(x - 2), or y = 6/23x - 104/23