Find dy/dx by implicit differentiation. arctan(4x^2y) = x + 5xy^2

Question

derivatives w inverse trig functions

Answer 1

Differentiate both sides of the equation using the chain rule.

 

(8xy + 4x²y') / (1 + 16x⁴y²) = 1 + 5y² + 10xyy'

 

 

Multiply both sides of the equation by the denominator on the left side.

 

8xy + 4x²y' = (1 + 16x⁴y²)(1 + 5y² + 10xyy')

 

8xy + 4x²y' = 1 + 5y² + 10xyy' + 16x⁴y²(1 + 5y² + 10xyy')

 

 

Complete these next steps on your own:

 

1) Distribute all terms to get rid of parentheses on the right side.

2) Move all y' terms to the left side.  Move all non-y' terms to the right side.

3) Factor out a y' on the left side.

4) Divide both sides of the equation by the coefficient of y'.

5) Finally obtain y'.

 

y' = dy/dx

 

Can you handle this?