How do you implicitly differentiate #7xy- 3 lny= 4/x#?

Answer 1

#dy/dx=-(7x^2y^2+4y)/(7x^3y-3x^2)#

Find the derivative of each part.

Use product rule:

#d/dx(7xy)=7y+7xy'#
Recall that #d/dx(lnu)=(u')/u# through the chain rule:
#d/dx(-3lny)=-(3y')/y#
Rewrite #4/x# as #4x^-1#.
#d/dx(4x^-1)=-4x^-2=-4/x^2#
Add all these derivatives to find the complete differentiated expression, then solve for #y'#.
#7y+7xy'-(3y')/y=-4/x^2#
Multiply everything by #x^2y# to clear fractions.
#7x^2y^2+7x^3y(y')-3x^2y'=-4y#
#y'(7x^3y-3x^2)=-7x^2y^2-4y#
#y'=-(7x^2y^2+4y)/(7x^3y-3x^2)=dy/dx#
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Answer 2

To implicitly differentiate 7xy3ln(y)=4x7xy - 3\ln(y) = \frac{4}{x}, follow these steps:

  1. Differentiate each term with respect to xx.
  2. Apply the chain rule when differentiating ln(y)\ln(y).

Differentiating each term:

ddx(7xy)ddx(3ln(y))=ddx(4x)\frac{d}{dx}(7xy) - \frac{d}{dx}(3\ln(y)) = \frac{d}{dx}\left(\frac{4}{x}\right)

Apply product rule to 7xy7xy:

7ddx(xy)+xddx(7y)3ddx(ln(y))=ddx(4x)7\frac{d}{dx}(xy) + x\frac{d}{dx}(7y) - 3\frac{d}{dx}(\ln(y)) = \frac{d}{dx}\left(\frac{4}{x}\right)

Apply chain rule to ln(y)\ln(y):

7(dydxy+xdydx)3(1ydydx)=ddx(4x)7\left(\frac{dy}{dx}y + x\frac{dy}{dx}\right) - 3\left(\frac{1}{y}\frac{dy}{dx}\right) = \frac{d}{dx}\left(\frac{4}{x}\right)

Differentiate 4x\frac{4}{x}:

4x2=4x2-\frac{4}{x^2} = -\frac{4}{x^2}

Simplify and solve for dydx\frac{dy}{dx}:

7xy+7y+xy3yy=4x27xy' + 7y + xy' - \frac{3}{y}y' = -\frac{4}{x^2}

7xy+xy3yy=4x27y7xy' + xy' - \frac{3}{y}y' = -\frac{4}{x^2} - 7y

xy(7+1)3yy=4x27yxy'(7 + 1) - \frac{3}{y}y' = -\frac{4}{x^2} - 7y

8xy3yy=4x27y8xy' - \frac{3}{y}y' = -\frac{4}{x^2} - 7y

y(8x3y)=4x27yy'(8x - \frac{3}{y}) = -\frac{4}{x^2} - 7y

y=4x27y8x3yy' = \frac{-\frac{4}{x^2} - 7y}{8x - \frac{3}{y}}

This is the implicit derivative of the given function.

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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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