Let f(x, y)=3 x^3 y^2 . Find (a) fx(x, y) (b) fy(x, y) (c) fx(1, y) (d) fx(x

step 1 Hello everyone here the function f is given by 3 x cube y square we need to find f a derivative

Let f(x, y)=3 x^3 y^2 . Find (a) fx(x, y) (b) ...

Let $f(x, y)=3 x^{3} y^{2} .$ Find $$ \begin{array}{lll}{\text { (a) } f_{x}(x, y)} & {\text { (b) } f_{y}(x, y)} & {\text { (c) } f_{x}(1, y)} \\ {\text { (d) } f_{x}(x, 1)} & {\text { (e) } f_{y}(1, y)} & {\text { (f) } f_{y}(x, 1)} \\ {\text { (g) } f_{x}(1,2)} & {\text { (h) } f_{y}(1,2)}\end{array} $$

Let $f(x, y)=3 x^{3} y^{2} .$ Find
$$
\begin{array}{lll}{\text { (a) } f_{x}(x, y)} & {\text { (b) } f_{y}(x, y)} & {\text { (c) } f_{x}(1, y)} \\ {\text { (d) } f_{x}(x, 1)} & {\text { (e) } f_{y}(1, y)} & {\text { (f) } f_{y}(x, 1)} \\ {\text { (g) } f_{x}(1,2)} & {\text { (h) } f_{y}(1,2)}\end{array}
$$

Key Concepts

Multivariable Functions

Multivariable functions are functions that depend on two or more independent variables. They are foundational in calculus and allow analysis of systems where more than one variable influences the output. In these functions, changes in individual input variables can be studied independently or jointly to understand the behavior of the function over its entire domain.

Partial Derivatives

Partial derivatives measure the rate of change of a multivariable function with respect to one variable at a time while keeping the other variables constant. This concept is critical in multivariable calculus as it extends the idea of differentiation to functions of several variables, enabling analysis of how changes in one dimension affect the function.

Power Rule in Differentiation

The power rule is a basic differentiation rule used in calculus which states that the derivative of x^n with respect to x is n*x^(n-1). When applied to multivariable functions, the rule is used on each variable independently while treating other variables as constants, which simplifies the process of finding partial derivatives.

Evaluation of Derivatives at Specific Points

Evaluating derivatives at specific points involves substituting given values into the derived expressions for the derivatives. This step is crucial for obtaining numerical insights into the behavior of the function at particular points in its domain, and is widely used in optimization and modeling to analyze local behavior of multivariable functions.

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