Let f(x, y, z)=x^3 y^5 z^7+x y^2+y^3 z . Find (a) fx y (b) fy z (c) fx z (d) f

step 1 Hello everyone here the function f is given by it is of the variable x y z which is x cube into

Let f(x, y, z)=x^3 y^5 z^7+x y^2+y^3 z . Find (a) fx y...

Let $f(x, y, z)=x^{3} y^{5} z^{7}+x y^{2}+y^{3} z .$ Find $$\begin{array}{lll}{\text { (a) } f_{x y}} & {\text { (b) } f_{y z}} & {\text { (c) } f_{x z}} & {\text { (d) } f_{z z}} \\ {\text { (e) } f_{z y y}} & {\text { (f) } f_{x x y}} & {\text { (g) } f_{z y x}} & {\text { (h) } f_{x x y z}}\end{array}$$

Let $f(x, y, z)=x^{3} y^{5} z^{7}+x y^{2}+y^{3} z .$ Find
$$\begin{array}{lll}{\text { (a) } f_{x y}} & {\text { (b) } f_{y z}} & {\text { (c) } f_{x z}} & {\text { (d) } f_{z z}} \\ {\text { (e) } f_{z y y}} & {\text { (f) } f_{x x y}} & {\text { (g) } f_{z y x}} & {\text { (h) } f_{x x y z}}\end{array}$$

Key Concepts

Partial Derivatives

This concept refers to the derivatives of multivariable functions taken with respect to one variable at a time while keeping the other variables constant. It forms the core idea of differentiating functions that depend on multiple variables, and it is essential for exploring how a function changes in one direction in its domain.

Mixed Partial Derivatives

Mixed partial derivatives involve taking the partial derivatives with respect to different variables in succession. They provide insight into how the change in one variable affects the rate of change with respect to another variable, and are critical in studying the curvature, optimization, and behavior of multivariable functions.

Higher Order Partial Derivatives

Higher order partial derivatives are obtained by differentiating a function more than once. When the differentiation involves different combinations and repeated variables, careful application of differentiation rules is needed. These derivatives help in understanding the intricacies of the function's behavior, such as concavity and interaction effects between variables.

Product Rule in Partial Differentiation

When a function involves products of functions of different variables, the product rule is applied for partial differentiation. This rule states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second, an essential tool when handling expressions that are multiplicative combinations of variable-dependent terms.

Clairaut's Theorem

Also known as the symmetry of second derivatives, Clairaut's theorem states that if the mixed partial derivatives of a function are continuous, then the order of differentiation does not matter. This property is useful in confirming the equality between certain mixed partial derivatives, under the assumption that the function is sufficiently smooth.

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