Implicit differentiation of xcosy+ycosx=1

[CACain]

Implicit Differentiation Problem -- Check my work?

I've worked it -- can someone just check my work?

Problem:

xcosy+ycos=1

My work:

[x (d/x)cosy + cosy (d/dx)x] + [y (d/dx)cosx + cosx (d/dx)y] = (d/dx) 1

-xsiny (dy/dx) + cos y - ysinx + cos x (dy/dx) = 0

-xsiny (dy/dx) + cos y = ysinx - cosy

dy/dx = (ysinx - cosy)/(-xsiny + cos x)


Meanwhile, could someone help me with this one...

squareroot (xy) = 1+(x^2)y

 

[kreil]

the first one looks good, as for the second:



(2)$$xy=1+x^2y$$

You are differentiating WRT y, which means that the derivative of x is 1, but the derivative of y is dy/dx. Make sure you use the product rule:

$$x\frac{dy}{dx}+y=x^2 \frac{dy}{dx}+2xy$$

$$x\frac{dy}{dx}-x^2 \frac{dy}{dx}=2xy-y$$

$$\frac{dy}{dx}(x-x^2)=2xy-y$$

$$\frac{dy}{dx}=\frac{2xy-y}{x-x^2}$$

~Josh

 

[HallsofIvy]

-xsiny (dy/dx) + cos y = ysinx - cosy

dy/dx = (ysinx - cosy)/(-xsiny + cos x)

You miswrote the first line above but obviously that was a typo since you got it right in the end. If I were your teacher I would prefer to see one more line:
-xsiny (dy/dx) + cos y = ysinx - cos x(dy/dx)

cos x (dy/dx)- x sin y (dy/dx)= (cos x- x sin y)(dy/dx)= y sin x- cos y

dy/dx = (ysinx - cosy)/(-xsiny + cos x)