How do you find #dy/dx# by implicit differentiation of #xy=4# and evaluate at point (-4,-1)?

Answer 1

#dy/dx=-1/4#

#xy=4#

Let's differentiate both sides

#xdy/dx+y=0#
#dy/dx=-y/x#
As, #y=4/x#
#dy/dx=-4/x^2#
At the point #(-4,-1)#
#dy/dx=-4/16=-1/4#
A tangent at #(-4,-1)# is
#y+1=-1/4(x+1)#
#y=-x/4-2#

graph{(y+2+x/4)(y-4/x)=0 [-17.28, 2.73, -4.94, 5.06]}

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Answer 2

To find ( \frac{dy}{dx} ) by implicit differentiation of ( xy = 4 ), follow these steps:

  1. Differentiate both sides of the equation with respect to ( x ).
  2. Use the product rule for differentiation on the left side.
  3. Solve the resulting equation for ( \frac{dy}{dx} ).
  4. Substitute the given point ( (-4, -1) ) into the expression for ( \frac{dy}{dx} ) to evaluate.

Starting with ( xy = 4 ):

  1. ( \frac{d}{dx}(xy) = \frac{d}{dx}(4) )
  2. Apply the product rule: ( y + x\frac{dy}{dx} = 0 )
  3. Solve for ( \frac{dy}{dx} ): ( \frac{dy}{dx} = -\frac{y}{x} )
  4. Substitute ( (-4, -1) ): ( \frac{dy}{dx} = -\frac{-1}{-4} = \frac{1}{4} )

So, ( \frac{dy}{dx} = \frac{1}{4} ) when evaluated at the point ( (-4, -1) ).

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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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