Tangent Space

In two-dimensional spaces, the tangent is line and it represents a linear equation approximately describing the change of $y$ relative to change of $x$ at that point

Image from [LibreTexts Mathematics](https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/14%3A_Differentiation_of_Functions_of_Several_Variables/14.04%3A_Tangent_Planes_and_Linear_Approximations#:~:text=Definition%3A tangent planes,-Let S be&text=z%3Df(x0%2C,(y−y0).)

In three-dimensional spaces, the equation of the tangent involves three variables, making it a tangent plane.

Partial Derivatives

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.

Given a function $f(x,y)$, the partial derivative of $f(x,y)$ with respect to $x$ is

$$ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h} $$

Notice how $y$ is a constant. Similarly

$$ \frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(x,y+h) - f(x,y)}{h} $$

Let’s try the calculating the partial derivatives for the function

$$ f(x,y) = x^2y + \sin y $$

First, let’s keep calculate the partial derivative relative to $x$ by keeping $y$ as a constant

$$ \begin{align} \frac{\partial f}{\partial x} &= \notag \frac{\partial }{\partial x} (x^2y + \sin y) \\ \notag &= \frac{\partial }{\partial x} (x^2y) + \frac{\partial }{\partial x}(\sin y) \\ \notag &= 2xy +0 \end{align} $$

• note: $\frac{\partial }{\partial x}(\sin y) = 0$ because we fix $y$ as a constant

Similarly, we calculate partial derivative relative to $y$ by keeping $x$ as a constant

$$ \begin{align} \frac{\partial f}{\partial y} &= \notag \frac{\partial }{\partial y} (x^2y + \sin y) \\ \notag &= \frac{\partial }{\partial y} (x^2y) + \frac{\partial }{\partial y}(\sin y) \\ \notag &= x^2 + \cos y \end{align} $$

Gradient