04. Partial Differentiation

4.1 Computation of Partial Derivatives

Partial Derivative

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a function of $n$ variables $x_1,x_2,\cdots ,x_n$
The partial derivative

$$\frac{\partial f}{\partial x_i}(p_1,\cdots ,p_n)$$

is the rate of change of $f$ at $(p_1,\cdots ,p_n)$ where we vary variable $x_i$ around $p_i$

Precisely the definition is

$$\frac{\partial f}{\partial x_i}(p_1,\cdots ,p_n)=\underset{h\to 0}{\lim }\frac{f(p_1,\cdots ,p_{i-1},p_i+h,p_{i+1},\cdots ,p_n)-f(p_1,\cdots ,p_n)}{h}$$

Note: finding partial derivatives of functions, just pretend the other variables are constants

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Full Derivatieves

$$\frac{df}{dx}\text{ is known as a full derivative}$$

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Partial Derivatives of single-value-functions

If $f(x)$ is a function of single variable then

$$\frac{df}{dx}=\frac{\partial f}{\partial x}$$

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Notation on Partial Derivatives

$\frac{\partial f}{\partial x}$ is also represented by $f_x$

If $f$ is a function of $x,y$ and we want to vary $y$ but keep $x$ constant then

$${\left(\frac{\partial f}{\partial y}\right)}_x$$

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Notation on Second Order Partial Derivatives

$$\frac{{\partial }^{2}f}{\partial x^2}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial x}\right)=f_{xx}$$ $$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y}\right)=f_{yx}$$ $$\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial x}\right)=f_{xy}$$ $$\frac{{\partial }^{2}f}{\partial y^2}=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial y}\right)=f_{yy}$$

Note that for continious derivatives then $f_{xy}=f_{yx}$

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4.2 The Chain Rule

Chain Rule

Let

$$F(t)=f(u(t),v(t))$$

where $u,v$ are differentiable and $f$ is continuously differentiable in $x,y$

Then

$$\frac{\partial F}{\partial t}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial t}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial t}$$

Non Examinable - page 25 :)

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Chain Rule in Multiple Variables corollary

Let $F(x,y)=f(u(x,y),v(x,y))$ with $u,v$ differentiable in each variable
with $f$ being continiously differentiable in each variable

$$\frac{\partial F}{\partial x}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}$$ $$\frac{\partial F}{\partial y}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y}$$

Proof

Follows from Chain Rule by treating first $x$ then $y$ as constants when differentiating

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4.3 Partial Differential Equations

Just examples, refer to Lecture Notes (page 29)