Differentiability

Equation of tangent line

Suppose $f$ is a single-variable function, and $f$ is differentiable at $x_0$. The equation of the tangent line to $f$ at $x_0$ is:

$$y - f(x_0) = f'(x_0)(x - x_0)$$

For 2-variable functions, they will have a tangent plane. For functions with more than 2 variables, they have a tangent space.

Differentiable

Suppose $z=f(x,y)$ where $f:D\rightarrow \mathbb{R}$; $D\subset \mathbb{R}^2$ and $(a,b)\in D$.

If $f_x(a,b)$ and $f_y(a,b)$ exists and $\Delta{z}$ can be expressed in the below form, then $f$ is differentiable at $(a,b)$.

$$\Delta z = \Delta x f_x(a,b) + \Delta y f_y(a,b) + \epsilon_1\Delta x + \epsilon_2 \Delta y$$

where $\epsilon_1$ and $\epsilon_2$ approach $0$ as $(\Delta x, \Delta y)$ approach $(0,0)$. OR

$f$ is differentiable at $(a,b)$ iff the limit exists:

$$\lim_{(x,y)\to(a,b)} \frac{f(x,y)-f(a,b)-f_x(a,b)\Delta x -f_y(a,b)\Delta y }{\sqrt{(x-a)^2+(y-b)^2}}$$

$f$ is said to be differentiable iff it is differentiable at every point in its domain. $$

If either $f_{x}$ or $f_y$ is non-existent at a point, then $f$ is not differentiable at that point.

If $f_x$ and $f_y$ are continuous throughout at an open region $D$, then $f$ is differentiable at every point of $D$.

The differentiability can also be proven by proving:

$$\lim_{\Delta \rho \to 0} \frac{\Delta z - \text{d}z}{\Delta \rho} = 0 \;\;\; \text{where} \;\;\; \Delta \rho = \sqrt{{\Delta x}^2 + {\Delta y}^2}$$

Implies Continuity

$$f \text{ is differentiable} \implies f \text{ is continous}$$

Derivative

Suppose $z=f(x,y)$ where $f$ is a differentiable function of $x$ and $y$.

$$\text{d}z = \frac{\partial z}{\partial x}\,\text{d}x + \frac{\partial z}{\partial y}\,\text{d}y =\Delta x f_x(a,b) + \Delta y f_y(a,b)$$

$\text{d}z$ is called the total differential of $f$.

Chain Rule

Let $x(t),y(t)$ be single-variable, differentiable functions and $f(x,y)$ be a 2-variable differentiable function having continuous first order partial derivatives.

$f\big(x(t),y(t)\big)$ is differentiable. $$

$$\frac{\text{d}f}{\text{d}t} = \frac{\partial f}{\partial x}\frac{\text{d}x}{\text{d}t} + \frac{\partial f}{\partial y}\frac{\text{d}y}{\text{d}t}$$

Can be extended for functions of more than 2 variables.