Posted by Melissa on Thursday, March 14, 2013 at 1:57pm.
Use implicit differentiation to find the slope of the tangent line to the curve y/( x + 4 y) = x^4 – 4
at the point (1,3/13)

Math  Steve, Thursday, March 14, 2013 at 3:17pm
y/(x+4y) = x^4 – 4
(y'(x+4y)  y(1+4y'))/(x+4y)^2 = 4x^3
(x+4y4y)y' = 4x^3 (x+4y)^2 +y
y' = 4x^2 (x+4y)^2 + y/x
y'(1) = 4(1)(112/13) + (3/13)/1 = 1/13

Math  Melissa, Thursday, March 14, 2013 at 3:27pm
Steve,
That still isn't the correct answer :(

Math  Steve, Thursday, March 14, 2013 at 4:00pm
Hmmm. Have you any clues? what is the correct answer? see any mistakes in my agebra?
If you go to wolframalpha.com and type in
derivative y/(x+4y) = x^4 – 4
you will see the value of y', which is what I got above. So, if my answer is wrong, I must have erred in my evaluation at (1, 3/13).

Math  Steve, Thursday, March 14, 2013 at 4:03pm
Ah. I see I didn't take (112/13)^2
You can fix that.

Math  Reiny, Thursday, March 14, 2013 at 4:12pm
I first changed the equation to:
y = x^5  4x + 4x^4y  16y
now differentiate ....
y' = 5x^4  4 + (4x^4)y' + 16x^3 y'  16y'
y'( 1  4x^4 + 16) = 5x^4  4 + 16x^3
using the point (1, 3/13)
y' (1416) = 5  4  48/13
19y' = 35/13
y' = 35/247

Math  Steve, Thursday, March 14, 2013 at 4:39pm
oops: 16x^3 y' ?

Math  Reiny, Thursday, March 14, 2013 at 6:06pm
yup, that's the one
should have been 16x^3y
should have realized there were too many y' hanging around
Thanks
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