Implicit Differentiation and Integration

[hoffenhelm]

Ok so I'm having problems on these couple of questions. Mostly the problem is when ln gets involved.

1. Find dy/dx
xlny+y=x^6y^6

2. (t+1/t)^2 dx |(1 to 2) <---- Integrating from 1 to 2

 

[skeeter]

d/dx[x*lny + y = x^6*y^6]

product rule in a couple of spots ... correct?

x*(1/y)(dy/dx) + lny + dy/dx = x^6*6y^5*(dy/dx) + y^6*6x^5

(x/y)(dy/dx) + dy/dx - 6x^6*y^5(dy/dx) = 6x^5*y^6 - lny

dy/dx[x/y + 1 - 6x^6*y^5] = 6x^5*y^6 - lny

dy/dx = [6x^5*y^6 - lny]/[x/y + 1 - 6x^6*y^5]

if you want to clear the fraction in the denominator, multiply numerator and denominator by y.


[t + (1/t)]^2 = t^2 + 2 + (1/t^2)

antiderivative will be ...

(t^3/3) + 2t - (1/t) you can now use the FTC and evaluate the definite integral with the given limits of integration.

 

[hoffenhelm]

Ok, thanks a lot for the help, just one quick question on the first part with clearing the fraction at the bottom. If I multiplied the numerator and denominator by y would I get:

dy/dx=[6x^5y^7 - y ln y]/[x-6x^6y^6 + y]

or

dy/dx=[6x^5y^7 - ln y]/[x - 6x^6y^6 +1]

not sure if I am doing it correctly