1. Find dy/dx for e^y = xy 2.If d/dx (f(x)) = g(x) and d/dx (g(x)) = f(3x), then d^2/dx^2 of (f(x^2)) is?

Question

I need help solving these problems. These are the answers. I can not fiugre out the inbetween steps. For #1 I believe you have to use implicit diff.
1. The answer is y/ (xy-x).
2. The answer is 4x^2 f (3x^2) = 2g(x^2)

Answer 1

1)

e^y = xy

To solve this one, we need to differentiate implicitly. Remember that the derivative of y is dy/dx. On the left hand side, we would use the chain rule, and on the right hand side, the product rule and then the chain rule.

Remember that the product rule goes, "The derivative of the first times the second, plus the derivative of the second times the first".

e^y = xy
(e^y)(dy/dx) = (1)y + (dy/dx)x

Next, you bring ALL terms attached to a (dy/dx) and bring it over to the left hand side.

(e^y)(dy/dx) - (dy/dx)x = y

Now, FACTOR dy/dx.

(dy/dx) [e^y - x] = y

And then, since we want to isolate (dy/dx), divide both sides by e^y - x.

(dy/dx) = y / [e^y - x]

For some reason I didn't quite get the same answer as you, unfortunately.

Answer 2

1. Find dy/dx for e^y = xy
e^y y' = xy' + y
y'(e^y -x) = y
y' = y/ (e^y -x)
if you substitute e^y= xy
then
y'= y/ (xy -x) .

2.If d/dx (f(x)) = g(x) and d/dx (g(x)) = f(3x), then d^2/dx^2 of (f(x^2)) is

d/dx (f(x^2)) = f'(x^2) 2x = 2x g(x^2)
d^2/dx^2 (f(x^2)) = 2x g'(x^2) 2x + 2g(x^2)
= 4x^2 f(3x^2) + 2g(x^2)

this time your answer is not right, maybe a typo.
.

Answer 3

check at addmath book