# implicit differentiation (finding the 1st and 2nd derivative)

• Oct 22nd 2009, 02:06 PM
driver327
implicit differentiation (finding the 1st and 2nd derivative)
With these, i cannot get the answer that is said from the book.
1. y^2 = e^(x^2) + 2x
I already got the 1st one which is: (xe^(x^2) + 1) / y

for the 2nd:
((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)(y')) / y^2

((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)((xe^(x^2) + 1) / y))
/ y^2

Multiplied by y and got:

(y^2*e^(x^2) ( 1+2x^2) - (xe^(x^2))(xe^(x^2) + 1)) / y^3 and this is

where I totally got lost... but the book said the answer is

(((2x^2)*(y^2) + y^2 - 2x)(e^(x^2)) - x^2e^(2x^2) - 1) / y^3

2. 2 *sqrt(y) = x - y

using the product rule:

I got y^(-1/2) = 1 - y'

I multiplied a sqrt(y) (to get rid of the fraction), but it didn't do me any good and eventually got

completely confused. The answer says sqrt(y)/(sqrt(y) + 1) for the 1st

and 1 / (2(sqrt(y) + 1) ^3 for the 2nd.
• Oct 22nd 2009, 02:46 PM
rn443
2sqrt(y) = x - y implies 4y = x^2 - 2xy + y^2. Differentiating yields 4y' = 2x - (2y + 2xy') + 2yy'. Use algebra to solve for y'.
• Oct 23rd 2009, 11:19 AM
driver327
Quote:

Originally Posted by rn443
2sqrt(y) = x - y implies 4y = x^2 - 2xy + y^2. Differentiating yields 4y' = 2x - (2y + 2xy') + 2yy'. Use algebra to solve for y'.

I am not exactly sure how you got 4y' from 2sqrt(y), but I tried this out again and got:

2((1/2) (y^(-1/2)) (y') = 1- y'

(y^(-1/2)) (y') = 1 - y'
y' + (y^(-1/2)) (y') = 1
y' (1+ (y^(-1/2))) = 1
y' = 1 / (1+ (y^(-1/2)))
Which i think amounts to:

y' = ((y^(1/2)) + 1)

What am I doing wrong here?
• Oct 23rd 2009, 11:38 AM
driver327
Quote:

Originally Posted by driver327
With these, i cannot get the answer that is said from the book.
1. y^2 = e^(x^2) + 2x
I already got the 1st one which is: (xe^(x^2) + 1) / y

for the 2nd:
((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)(y')) / y^2

((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)((xe^(x^2) + 1) / y))
/ y^2

Multiplied by y and got:

(y^2*e^(x^2) ( 1+2x^2) - (xe^(x^2) + 1)(xe^(x^2) + 1)) / y^3 and this

is where I totally got lost... but the book said the answer is

(((2x^2)*(y^2) + y^2 - 2x)(e^(x^2)) - x^2e^(2x^2) - 1) / y^3

I realized that i dropped the 1 (in blue) when i multiplied the y so i now get:

(y^2*e^(x^2) ( 1+2x^2) - (xe^(x^2) + 1)^2) / y^3
• Oct 24th 2009, 08:02 AM
driver327
really need help on this 1...