Question

Use implicit differentiation to find dy dx for x ≥ −7. y x 7 = xy 9

Answer 1

So, using implicit differentiation to find dy/dx for x ≥ −7. yx⁷ = xy⁹, dy/dx = (y⁹ + 7x⁶y)/(x⁷ - 9xy⁸)

To use implicit differentiation to find dy/dx for x ≥ −7. yx⁷ = xy⁹, we proceed as follows

Since we have yx⁷ = xy⁹, using the product rule on both sides of the equation.

duv/dx = udv/dx + vdu/dx

In yx⁷, u = y and v = x⁷

du/dx = dy/dx and dv/dx = 7x⁶

du'v'/dx = u'dv'/dx + v'du'/dx

In xy⁹, u' = x and v' = y⁹

du'/dx = 1 and dv'/dx = 9y⁸dy/dx

So, dyx⁷/dx = dxy⁹/dx

duv/dx = du'v'/dx

udv/dx + vdu/dx = u'dv'/dx + v'du'/dx

y × 7x⁶ + x⁷dy/dx = x × 9y⁸dy/dx + y⁹(1)

7x⁶y + x⁷dy/dx = 9xy⁸dy/dx + y⁹

Collecting similar terms, we have that

x⁷dy/dx - 9xy⁸dy/dx = y⁹ + 7x⁶y

Factorizing out dy/dx, we have that

(x⁷ - 9xy⁸)dy/dx = y⁹ + 7x⁶y

dy/dx = (y⁹ + 7x⁶y)/(x⁷ - 9xy⁸)

So, dy/dx = (y⁹ + 7x⁶y)/(x⁷ - 9xy⁸)