# What are the first, second, and third order partial derivatives of #f(x,y,z)=ln(xyx)#?

Supposing it is

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The first-order partial derivatives are:

[\frac{{\partial f}}{{\partial x}} = \frac{1}{{x}}, \quad \frac{{\partial f}}{{\partial y}} = \frac{1}{{y}}, \quad \frac{{\partial f}}{{\partial z}} = 0]

The second-order partial derivatives are:

[\frac{{\partial^2 f}}{{\partial x^2}} = -\frac{1}{{x^2}}, \quad \frac{{\partial^2 f}}{{\partial y^2}} = -\frac{1}{{y^2}}, \quad \frac{{\partial^2 f}}{{\partial z^2}} = 0]

The mixed partial derivatives are:

[\frac{{\partial^2 f}}{{\partial x \partial y}} = \frac{{\partial^2 f}}{{\partial y \partial x}} = -\frac{1}{{xy}}] [\frac{{\partial^2 f}}{{\partial x \partial z}} = \frac{{\partial^2 f}}{{\partial z \partial x}} = 0] [\frac{{\partial^2 f}}{{\partial y \partial z}} = \frac{{\partial^2 f}}{{\partial z \partial y}} = 0]

The third-order partial derivatives are:

[\frac{{\partial^3 f}}{{\partial x^3}} = \frac{{2}}{{x^3}}, \quad \frac{{\partial^3 f}}{{\partial y^3}} = \frac{{2}}{{y^3}}, \quad \frac{{\partial^3 f}}{{\partial z^3}} = 0]

The mixed third-order partial derivatives are:

[\frac{{\partial^3 f}}{{\partial x^2 \partial y}} = \frac{{\partial^3 f}}{{\partial x \partial y^2}} = \frac{{2}}{{xy^2}}] [\frac{{\partial^3 f}}{{\partial x^2 \partial z}} = \frac{{\partial^3 f}}{{\partial x \partial z^2}} = 0] [\frac{{\partial^3 f}}{{\partial y^2 \partial z}} = \frac{{\partial^3 f}}{{\partial y \partial z^2}} = 0]

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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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