class: center, middle, inverse, title-slide .title[ # Partial Derivatives ] .subtitle[ ## Session 02 ] .author[ ### Alejandro Ucan ] .date[ ### 2024-02-10 ] --- # Goals of the Session * We will introduce the concept of partial derivatives. <br/><br/> * We will apply the general rules of partial derivatives. <br/><br/> --- # Motivation The **heat index** ($I$) measures the combined effects of temperature and humidity. `\(I\)` measures the sensation of air temperature when the temperature is `\(T\)` and the relative humidity is `\(H.\)`  -- What is the rate of change of `\(I\)` when we change `\(T\)` at the point `\((96,70)\)`? --- # Partial Derivative > __Definition:__ Given a function `\(f:\mathbb{R}^n\to \mathbb{R},\)` the partial derivative with respect to the variable `\(x_j,\)` denoted by `\(f_{x_j}=\frac{\partial f}{\partial x_j},\)` is defined as the limit `$$\lim_{h\to 0} \frac{f(x_1,\cdots,x_j+h,x_{j+1},\cdots, x_n)-f(x_1,\cdots,x_j,x_{j+1},\cdots, x_n)}{h}.$$` -- #### Example 1: Compute the partial derivative with respect to `\(x\)` of the function `\(f(x,y)=3xy.\)` -- `$$f_x(x,y)=\lim_{h\to 0}\frac{3(x+h)y-3xy}{h}=\lim_{h\to 0}\frac{3(x+h-x)y}{h}=\lim_{h\to 0}\frac{3yh}{h}=3y.$$` --- #### Ejemplo 3: Compute the partial derivative with respect to `\(y\)` of the function `\(f(x,y)=xy-zx+4y.\)` -- <br/><br/> `$$f_y(x,y,z)=\lim_{h\to 0}\frac{x(y+h)-zx+4(y+h)-(xy-zx+4y)}{h}=\lim_{h\to 0}\frac{xh+4h}{h}=x+4.$$` -- <br/><br/> respect to `\(z\)`? -- <br/><br/> `$$f_y(x,y,z)=\lim_{h\to 0}\frac{xy-(z+h)x+4y-(xy-zx+4y)}{h}=\lim_{h\to 0}\frac{hx}{h}=x.$$` --- ## How to Partially Derive? > To partially derive a function with respect to a variable, we just derive the function assuming that the other variables are constants. -- #### Example 4: Compute the partial derivatives `\(f_x,\,f_y\)` y `\(f_z\)` for the next functions: <br/><br/> * `\(f(x,y,z)=\frac{\sin(x)}{z}+x\cos(y)\)` * `\(f(x,y,z)=z^2e^{x+y}\)` * `\(f(x,y,z)=x\ln(z)+y\sqrt{x}.\)` --- # Geometric Interpretation of Partial Derivatives Hereby we present the graph of the function `\(f(x,y)=x^2+xy+y^3\)` at the point `\((1,1).\)` <iframe src="https://www.geogebra.org/calculator/aemfszpg?embed" width="1200" height="400" allowfullscreen style="border: 1px solid #e4e4e4;border-radius: 4px;" frameborder="0"></iframe> --- # Modelling with Partial Derivatives: > The Argentine inflation (in 2019) is given by the function `\(h(x,y)=e^{x+y}\sqrt{4-3x^2-2y^2},\)` where `\(h\)` is measured in percentage, and `\((x,y)\)` represents a point in the city. If we observe at the point `\((1,1)\)`, what is the rate of change of inflation when we move in the vertical direction? And in the horizontal direction? -- `$$\frac{\partial f}{\partial x}=-e^{x + y} \frac{-4 + 3 x + 3 x^2 + 2 y^2}{\sqrt{4 - 3 x^2 - 2 y^2}}$$` `$$\frac{\partial f}{\partial x}=-e^{x + y} \frac{3 x^2 + 2 (-2 + y + y^2)}{\sqrt{4 - 3 x^2 - 2 y^2}}$$` --- # Implicit Partial Derivation #### Example 5: Assume that `\(x,y,z\)` satisfy the following equation: `$$x^2+y^2-z^2=0.$$` Compute `\(z_x\)` and `\(z_y.\)` -- <br/><br/> We have that `$$\frac{\partial z}{\partial x}=\frac{x}{z}$$` `$$\frac{\partial z}{\partial y}=\frac{y}{z}.$$` --- # Properties of Partial Derivatives > __Theorem:__ Let `\(f,g:\mathbb{R}^n\to\mathbb{R}\)` two function where `\(n=2,3.\)` The following are valid: <br/><br/> * `\((f\pm g)_x=f_x\pm g_x.\)` <br/><br/> * `\((fg)_x = (f_x)g + f(g_x).\)` <br/><br/> * If `\(g\neq 0,\)` then `\((f/g)_x =\frac{(f_x)g-f(g_x)}{g^2}.\)`