Let {eq}f(x, y, z) = x^2 y^3 z^4 {/eq}. Find {eq}\displaystyle \frac {\partial^3 f}{ \partial x \partial y \partial z} (4, 3, 2). {/eq}

Question:

Partial Derivatives:

Let {eq}u = f(x, y, z), {/eq}

where {eq}x, y, z {/eq} are independent variable and {eq}u {/eq} is dependent variable.

Now, we want to differentiate {eq}u {/eq} with respect to {eq}x, y {/eq} and {eq}z, {/eq} then we introduce a special symbol {eq}\partial, {/eq} that is called partial differential operator.

Since, there are three independent variables {eq}(x, y, z) {/eq} , so there exists three partial derivatives, i.e. {eq}\frac{{\partial f}}{{\partial x}}, \frac{{\partial f}}{{\partial y}} {/eq} and {eq}\frac{{\partial f}}{{\partial z}}. {/eq}

{eq}\frac{{\partial f}}{{\partial x}} {/eq} means "partial derivative of {eq}f {/eq} with respect to {eq}x {/eq} treating {eq}y, z {/eq} as a constant",

{eq}\frac{{\partial f}}{{\partial y}} {/eq} means "partial derivative of {eq}f {/eq} with respect to {eq}y {/eq} treating {eq}z, x {/eq} as a constant".

{eq}\frac{{\partial f}}{{\partial z}} {/eq} means "partial derivative of {eq}f {/eq} with respect to {eq}z {/eq} treating {eq}x, y {/eq} as a constant".

We recall the following formula of differentiation:

{eq}\frac{\partial }{{\partial x}}\left( {{x^n}} \right) = n{x^{n - 1}}. {/eq}

Answer and Explanation: 1