Use implicit differentiation to find the derivative y' of the following: Wrote the solution of each number

Question

Use implicit differentiation to find the derivative y' of the following: Wrote the solution of each number

6.y2e2x + xy3 - 1=0

ans. y' =y2 + 2ye2x/ 2e2x + 3xy

Answer 1

Using implicit differentiation on each term:

y²e^2x + xy³ - 1= 0

The first term is the product rule d/dx f g = f g' + f' g

so the first is y² and the second is e^2x. You must also use the chain rule.

So its derivative with respect to x is

y² ( e^2x(2)) + 2 y y' (e^2x)

The second term is also a product rule.

x (3y² y') + (1) (y³)

The third term is a constant so its derivative is 0.

Putting the pieces together

y² ( e^2x(2)) + 2 y y' (e^2x) + x (3y² y') + (1) (y³) + 0 = 0

Now clean up each term .

2y²e^2x + 2 y e^2x y' + 3 x y² y' + y³ = 0

Now put the terms that do NOT contain a y' on the right side of the =

2 y e^2x y' + 3 x y² y' = -2y²e^2x - y³

Now factor the y' on the left side. Then divide. Then simplify.