How do you differentiate #xy = cot(xy)#?

Answer 1

#dy/dx= ( ycsc^2(xy) - y)/(x - xcsc^2(xy))#

Start off by differentiating both sides.

#d/dx (xy) = d/dx (cot(xy))#

First, let's discuss how to differentiate the left side. We can apply the product rule.

#x(d/dx y) + y(d/dxx) = d/dx (cot(xy))#
#d/dx y = dy/dx# due to implicit differentiation and the chain rule.

So:

#x(d/dx y) + y(d/dxx) = d/dx (cot(xy))#
#xdy/dx + y = d/dx (cot(xy))#

Next, we can differentiate the right side using the chain rule. In short terms, the chain rule is:

  1. The derivative of the outer function, with the inner function plugged in...
  2. ...multiplied by the derivative of the inner function.
The outer function is #cot(x)#. The inner function is #xy#.
The derivative of the outer function is #csc^2(x)#. Plug in the inner function and we get #csc^2(xy)#. Then we multiply this by the derivative of the inner function.
So #xdy/dx + y = d/dx (cot(xy))# becomes:
#xdy/dx + y = csc^2(xy) * d/dx (xy)#
We've already solved #d/dx (xy)# on the left side, so we can just copy it:
#xdy/dx + y = csc^2(xy) * [xdy/dx + y]#
#xdy/dx + y =xcsc^2(xy) dy/dx + ycsc^2(xy) #
Then we need to isolate #dy/dx# so we bring all #dy/dx# terms to one side:
#xdy/dx - xcsc^2(xy) dy/dx = ycsc^2(xy) - y#
Then we factor out #dy/dx# to isolate it:
#dy/dx(x - xcsc^2(xy))= ycsc^2(xy) - y#
#dy/dx= ( ycsc^2(xy) - y)/(x - xcsc^2(xy))#
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Answer 2

To differentiate xy = cot(xy), we'll use implicit differentiation.

The derivative of xy with respect to x is y + x(dy/dx).

The derivative of cot(xy) with respect to x is -csc^2(xy) * (y + x(dy/dx)).

Now, equating the derivatives, we get:

y + x(dy/dx) = -csc^2(xy) * (y + x(dy/dx)).

Solving for dy/dx, we get:

dy/dx = - (y + x(dy/dx)) / (x + y*csc^2(xy)).

This is the derivative of xy = cot(xy) with respect to x.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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