PX153 - G1 - partital differentiation

we need to extend calculus to cover multivariate functions
the derivative of a function of one variable is defined by:

d f \frac{x}{d x} = lim_{Δ x \to \infty} \frac{f (x + Δ x) - f (x)}{Δ x}

for a function of two variables, we define the partial derivative:

\frac{\partial f (x, y)}{\partial x} = lim_{Δ x \to \infty} \frac{f (x + Δ x, y) - f (x, y)}{Δ x} = (\frac{\partial f (x, y)}{\partial x})_{y}

- the subscript $y$ implies constant $y$

similarly,

\frac{\partial f (x, y)}{\partial y} = lim_{Δ y \to \infty} \frac{f (x, y + Δ y) - f (x, y)}{Δ y} = (\frac{\partial f (x, y)}{\partial y})_{x}

we don't always specify what is held constant, but it is important that you're always clear about what is held constant
for multiple variables, $x_{i}$ :

\frac{\partial}{\partial x_{i}} f (x_{1}, . . . x_{i}, . . . x_{N}) = lim_{Δ x_{i} \to \infty} \frac{f (x_{1}, . . . x_{i} + Δ x_{i}, . . . x_{N}) - f x_{1}, . . . x_{i}, . . . x_{N}}{Δ x_{i}}

eg: $f (x, y) = e^{x y^{2}} + \frac{y}{2}$
- $f_{x} = y^{2} e^{x} y^{2}$ and $f_{y} = 2 y x e^{x y^{2}} + \frac{1}{2}$
eg: $f (x, y) = e^{x} \cos y + 6 x + 3$
- $f_{x} = e^{x} \cos y + 6$ and $f_{y} = - e^{x} \sin y$
similarly, we can define higher order derivatives
eg:

\frac{\partial}{\partial x \partial y} f (x, y) = \frac{\partial}{\partial x} (\frac{\partial f}{\partial y}) = lim_{Δ \to \infty} \frac{\frac{\partial f}{\partial y} (x - Δ x, y) - \frac{\partial f}{\partial y} (x, y)}{\partial y \partial x}

eg: to find all first and second order derivatives of: $f (x, y) = x^{3} y^{2} + y e^{x} + 6 y^{3} + 1$ $\frac{\partial f}{\partial x} = 3 x^{2} y^{2} + y e^{x}$ $\frac{\partial f}{\partial y} = 2 x^{3} y + e^{x} + 18 y^{2}$ $\frac{\partial^{2} f}{\partial x^{2}} = 6 x y^{2} + y e^{x}$ $\frac{\partial^{2} f}{\partial y^{2}} = 2 x^{3} + 36 y$

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