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The method of calculating derivatives from first principles is about getting back to the basics. This approach is foundational for understanding how derivatives, including partial derivatives, truly work. The aim is to comprehend how a function changes with a very small shift in one of its input variables, while maintaining the other inputs constant. In the case of partial derivatives like \( \frac{\partial f}{\partial x} \), you focus on observing infinitesimal changes in the \(x\)-component while keeping \(y\) fixed. It's like observing a reaction to a change if you slightly nudge one aspect while holding others still.

To apply first principles to partial derivatives, consider the specific point in the domain where you're evaluating this change. Use the concept of limits to determine how the function behaves as this shift becomes infinitely small.

The limit definition of a derivative is a key mathematical tool. It enables us to precisely determine the slope or rate of change at a specific point. When considering the limit definition for partial derivatives, it involves evaluating how the function changes as one input variable approaches zero while others are constant.

For example, to find \( \frac{\partial f}{\partial x} \), you would consider:

• \( \lim_{h \to 0} \frac{f(a + h, b) - f(a, b)}{h} \)

This expression assesses the change in the function when \(x\) is increased by a tiny amount \(h\) while \(y\) stays at \(b\). Similarly, to determine \( \frac{\partial f}{\partial y} \), you explore:

• \( \lim_{k \to 0} \frac{f(a, b + k) - f(a, b)}{k} \)

Here, the function's change is observed when \(y\) is tweaked by \(k\), with \(x\) constant.

Derivative evaluation is the phase where all calculations culminate. You take what you understand about partial derivatives and compute specific values. You plug in your expressions derived from the limit definitions into your known derivatives, simplifying and calculating precisely.

For instance, in the example given with \( f(x, y) = 2x^2 - xy + y^2 \), after determining that \( \frac{\partial f}{\partial x} = 4x - y \) and \( \frac{\partial f}{\partial y} = -x + 2y \), evaluation involves substituting specific values.

• For the point \((1, 2)\), evaluate \(4(1) - 2 = 2\)
• For \( -1 + 4 \), you find \(3\)

These values represent the slope of the function surface in the respective directions at \((1, 2)\).

Function expansion refers to expressing a function at a shifted point in terms of its original point. This concept is crucial for finding derivatives using first principles, as it involves symmetric changes exerted on individual inputs while keeping others unaffected.

In the context of determining \( \frac{\partial f}{\partial x} \), when given the function \(f(x, y)\), you replace \(x\) with \(x + h\) in the whole function expression. Expanding this involves:

• \( f(x + h, y) = 2(x + h)^2 - (x + h)y + y^2 \)
• Breaking down: \( = 2x^2 + 4xh + 2h^2 - xy - hy + y^2 \)
• Simplifying makes it easier to find the limit as \(h\) approaches zero

Similarly, function expansion for \( \frac{\partial f}{\partial y} \) involves replacing \(y\) with \(y + k\) and expanding accordingly.