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In multivariable calculus, a partial derivative measures how a function changes as one of its variables changes, keeping all other variables constant. This is similar to a regular derivative, but applied to functions of more than one variable.
Imagine you have a function, like the surface of a landscape. Each direction you move could change your elevation - going north or south might increase or decrease your height separately from moving east or west.

• Notation: The partial derivative with respect to a variable like \(x\) is noted as \(\frac{\partial f}{\partial x}\) or \(f_x\).
• Calculation: To find \(f_x(x, y)\) for a function of two variables \(f(x, y)\), treat \(y\) as a constant and differentiate with respect to \(x\).
• Purpose: Used to see how the function's rate of change at any given point changes in a specific direction.

For the function \(f(x, y) = \sin x \cdot \cos y\), we found the partial derivative with respect to \(x\) to be \(f_x = \cos x \cdot \cos y\). We do the same for \(y\), resulting in \(f_y = -\sin x \cdot \sin y\).
Evaluating at the point \((0,0)\) gives us a clearer idea of the rates of change there.

Multivariable calculus extends the concepts of single-variable calculus to functions of two or more variables. It's like adding more dimensions to the calculus you're already familiar with.
A key tool in multivariable calculus is the concept of the gradient, a vector that summarizes the rates of change in all directions and plays a central role in defining the tangent plane.

• Functions of Multiple Variables: These are surfaces or higher-dimensional analogs rather than curves. For example, \(f(x, y)\) represents a surface where each point \((x, y)\) has a corresponding output \(z\).
• Tangent Plane: Just as a tangent line touches a curve at one point, a tangent plane touches a surface at a point, providing the best linear approximation of the surface at that point.
• Applications: Used in many fields, from physics to economics, where phenomena depend on multiple inputs.

Understanding multivariable calculus allows us to deal with more complex real-world problems where multiple factors influence the outcome. The tangent plane equation from the exercise, \(z = x\), appears from a flat plane touching the surface at one point.

In calculus, the product rule is an essential technique used when taking the derivative of a product of two functions. This rule is especially useful in scenarios involving functions of several variables.
The product rule states that if you have two functions, \(u\) and \(v\), then the derivative of their product is: \[\frac{d}{dx}(u \cdot v) = u'v + uv'\] In the context of multivariable functions, like in our problem:

• Using the Product Rule: For \(f(x, y) = \sin x \cdot \cos y\), with \(u = \sin x\) and \(v = \cos y\), use the product rule to find \(f_x\).
• Step-by-Step: Derive \(\sin x\) while treating \(\cos y\) as constant to find \(f_x\).
• Result: Applying this rule simplifies finding how the function's value changes as \(x\) changes, essential for formulating the tangent plane.

Mastering the product rule allows you to differentiate more complex expressions with ease. It expands your toolkit for tackling problems in calculus involving products of functions. In our example, it helped yield \(f_x = \cos x \cdot \cos y\), crucial for the tangent plane's determination.