Find dy/dx by implicit differentiation

Question

tan³(xy²+y) = x

Answer 1

Eric,

The easiest way that I see for attacking this is by multiple substitution.

 

tan³(xy² + y) = x  can be rewritten as u³ = x, where u = tan(w),

and w = xy² + y.  From here we will need to know du and dw.

 

du = sec²(w)dw          dw = y² + 2xydy + dy

 

The derivative of u³ = x is:

 

3u²du = dx       Now substitute all the pieces back in.

 

3tan²(xy² + y)sec²(xy² + y)(y² + 2xydy + dy) = dx

 

Divide out all the factors that do not include dy:

 

y² + 2xydy + dy = dx/[3tan²(xy² + y)sec²(xy² + y)]

 

Subtract any terms which do not include dy and factor out dy:

 

dy(2xy + 1) = dx/[3tan²(xy² + y)sec²(xy2 + y)] - y²

 

Divide by dx and by (2xy + 1):

 

dy/dx = 1/[3tan²(xy² + y)sec²(xy2 + y)(2xy + 1)] - y²/(2xy + 1)

 

I hope that helps.