# How do you find dz/dx and dz/dy of #ln(xy+yz+xz) = 5# if x, y, and z are all positive?

Also, since we have the information that all variables are always positive, the expression is well defined (recall that logarithms are only defined for positive numbers).

Applying the chain rule to the left side of the equation, we have:

Doing the same, now with respect to y:

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To find ( \frac{dz}{dx} ) and ( \frac{dz}{dy} ) of ( \ln(xy + yz + xz) = 5 ), differentiate both sides of the equation with respect to ( x ) and ( y ) separately, treating ( z ) as a function of ( x ) and ( y ).

For ( \frac{dz}{dx} ):

- Differentiate both sides of the equation with respect to ( x ) using the chain rule.
- Obtain ( \frac{d}{dx} \ln(xy + yz + xz) = \frac{d}{dx} 5 ).
- Simplify and solve for ( \frac{dz}{dx} ).

For ( \frac{dz}{dy} ):

- Differentiate both sides of the equation with respect to ( y ) using the chain rule.
- Obtain ( \frac{d}{dy} \ln(xy + yz + xz) = \frac{d}{dy} 5 ).
- Simplify and solve for ( \frac{dz}{dy} ).

Note: In both cases, ( z ) is treated as a function of ( x ) and ( y ), so you'll need to use the chain rule to differentiate the natural logarithm term.

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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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