Question

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

Answer 1

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

Explanation:

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`