Question

What is the derivative of `e^y cos x = 3 + sin(xy)`?

Answer 1

`dy/dx=(e^ysinx+ycos(xy))/(e^ycosx-xcos(xy))`

Explanation:

`"differentiate " e^ycosx" using "color(blue)"product rule"`

`"Given " f(x)=g(x)h(x)" then "`

`f'(x)=g(x)h'(x)+h(x)g'(x)`

`"differentiate " sin(xy)" using "color(blue)" chain rule"`

`"Given " f(x)=g(h(x))" then"`

`f'(x)=g'(h(x)).h'(x)`

`"differentiate "color(blue)"implicitly with respect to x"`

`rArre^y(-sinx)+cosxe^y.dy/dx=cos(xy).d/dx(xy)`

`rArre^ycosx.dy/dx-e^ysinx=cos(xy)[x.dy/dx+y]`

`rArre^ycosxdy/dx-xcos(xy)dy/dx=ycos(xy)+e^ysinx`

`rArr dy/dx(e^ycosx-xcos(xy))=e^ysinx+ycos(xy)`

`rArr dy/dx=(e^ysinx+ycos(xy))/(e^ycosx-xcos(xy))`