Solve by using implicit differentiation

Question

a. x^9y+xy^7 =6x

b. 9x^2+y^2=9, (0,3)

Answer 1

Problem A

x⁹y + xy⁷ = 6x

9x⁸y + x⁹(dy/dx) + y⁷ + 7xy⁶(dy/dx) = 6 <-- Use Product Rule and let y' = dy/dx

(x⁹+7xy⁶)(dy/dx) = 6 - 9x⁸y - y⁷

dy/dx = (6-9x⁸y-y⁷)/(x⁹+7xy⁶)

Problem B

9x²+y² = 9

18x+2y(dy/dx) = 0

2y(dy/dx) = -18x

dy/dx = -9x/y

At (0,3) --> dy/dx = -9(0)/3 = 0/3 = 0

Remember that in implicit differentiation, y is treated as a constant and its derivative is dy/dx.

Hope the answers and explanations helped!

Answer 2

a. x^9y+xy^7 =6x

b. 9x^2+y^2=9, (0,3)

Hello Janet. To start I like taking these types of questions one term at a time. lets do a first...

remember that we treat y as a function of x in implicit differentiation. We use the chain rule here. Lets look at the first term...

dy/dx of x⁹ Y uses the product rule. That is, the derivative of the first times the second term, plus the second term times the derivative of the first, or, 9x⁸Y+x⁹y'

The second term is 1*y⁷+ 7xy⁶y'

on the right hand side we differentiate to get 6

so as a whole we have. 9x⁸Y+x⁹y' + y⁷+ 7xy⁶y' = 6

Then solve for y'

y' = (6 - 9x⁸y- y⁷)/(x⁹+7xy⁶)

For b) solve in the same way and plut in the falues for x and y. We get: y' = -9x/y. evaluated at (0,3) we get y' = 0