Find dy/dx by implicit differentiation

Question

a)x³ + xtan ^-1 y = ey

 

 

b)sin ^-1 (xy) = cos ^-1 (x - y)

 

 

 

 

Please, do them step by step

Answer 1

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Answer 2

a)    x^³ + xtan ^-1 y = e^y

 

3x² + tan^-1y + x[1/(1+y²)]y' = y'e^y  is the first step,

 

 

using implicit differentiation w.r.t. x & assuming y to be some function of x.

 

Next step: Rearrange so that all terms not involving factor y' are on one side of the equation,

and those involving y' are on the other side, and factor out y' and then finally solving for y'.

Take care with all signs of terms (sign changes when a term moves to the opposite of its former side).

Your y' solution will be in terms of x and also y.

 

Similar techniques apply to the second problem.