# What is the significance of partial derivative? Give an example and help me to understand in brief.

Feb 21, 2018

See below.

#### Explanation:

I hope it helps.

The partial derivative is intrinsically associated to the total variation.

Suppose we have a function $f \left(x , y\right)$ and we want to know how much it varies when we introduce an increment to each variable.

Fixing ideas, making $f \left(x , y\right) = k x y$ we want to know how much it is

$\mathrm{df} \left(x , y\right) = f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) - f \left(x , y\right)$

In our function-example we have

$f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) = k \left(x + \mathrm{dx}\right) \left(y + \mathrm{dy}\right) = k x y + k x \mathrm{dx} + k y \mathrm{dy} + k \mathrm{dx} \mathrm{dy}$

and then

$\mathrm{df} \left(x , y\right) = k x y + k x \mathrm{dx} + k y \mathrm{dy} + k \mathrm{dx} \mathrm{dy} - k x y = k x \mathrm{dx} + k y \mathrm{dy} + k \mathrm{dx} \mathrm{dy}$

Choosing $\mathrm{dx} , \mathrm{dy}$ arbitrarily small then $\mathrm{dx} \mathrm{dy} \approx 0$ and then

$\mathrm{df} \left(x , y\right) = k x \mathrm{dx} + k y \mathrm{dy}$

but generally

$\mathrm{df} \left(x , y\right) = f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) - f \left(x , y\right) = \frac{1}{2} \left(2 f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) - 2 f \left(x , y\right) + f \left(x + \mathrm{dx} , y\right) - f \left(x + \mathrm{dx} , y\right) + f \left(x , y + \mathrm{dy}\right) - f \left(x , y + \mathrm{dy}\right)\right) =$

$= \frac{1}{2} \frac{f \left(x + \mathrm{dx} , y\right) - f \left(x , y\right)}{\mathrm{dx}} \mathrm{dx} + \frac{1}{2} \frac{f \left(x , y + \mathrm{dy}\right) - f \left(x , y\right)}{\mathrm{dy}} \mathrm{dy} +$

$+ \frac{1}{2} \frac{f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) - f \left(x , y + \mathrm{dy}\right)}{\mathrm{dx}} \mathrm{dx} + \frac{1}{2} \frac{f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) - f \left(x + \mathrm{dx} , y\right)}{\mathrm{dy}} \mathrm{dy}$

now making $\mathrm{dx} , \mathrm{dy}$ arbitrarily small we have

$\mathrm{df} \left(x , y\right) = \frac{1}{2} \left(2 {f}_{x} \left(x , y\right) \mathrm{dx} + 2 {f}_{y} \left(x , y\right) \mathrm{dy}\right) = {f}_{x} \left(x , y\right) \mathrm{dx} + {f}_{y} \left(x , y\right) \mathrm{dy}$

so we can compute the total variation for a given function, by calculating the partial derivatives ${f}_{{x}_{1}} , {f}_{{x}_{2}} , \cdots , {f}_{{x}_{n}}$ and compounding

$\mathrm{df} \left({x}_{1} , {x}_{2} , \cdots , {x}_{n}\right) = {f}_{{x}_{1}} {\mathrm{dx}}_{1} + \cdots + {f}_{{x}_{n}} {\mathrm{dx}}_{n}$

Here, the quantities ${f}_{{x}_{i}}$ are called partial derivatives and can also be represented as

$\frac{\partial f}{\partial {x}_{i}}$

In our example

${f}_{x} = \frac{\partial f}{\partial x} = k x$ and

${f}_{y} = \frac{\partial f}{\partial y} = k y$

NOTE

${f}_{x} \left(x , y\right) = {\lim}_{\begin{matrix}\mathrm{dx} \to 0 \\ \mathrm{dy} \to 0\end{matrix}} \frac{f \left(x + \mathrm{dx} , y\right) - f \left(x , y\right)}{\mathrm{dx}} = {\lim}_{\begin{matrix}\mathrm{dx} \to 0 \\ \mathrm{dy} \to 0\end{matrix}} \frac{f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) - f \left(x , y\right)}{\mathrm{dx}}$

${f}_{y} \left(x , y\right) = {\lim}_{\begin{matrix}\mathrm{dx} \to 0 \\ \mathrm{dy} \to 0\end{matrix}} \frac{f \left(x , y + \mathrm{dy}\right) - f \left(x , y\right)}{\mathrm{dy}} = {\lim}_{\begin{matrix}\mathrm{dx} \to 0 \\ \mathrm{dy} \to 0\end{matrix}} \frac{f \left(x + \mathrm{dx} , y + \mathrm{dy}\right) - f \left(x , y\right)}{\mathrm{dy}}$

Feb 22, 2018

See below.

#### Explanation:

To supplement Cesareo's answer above, I will provide a less mathematically rigorous introductory definition.

The partial derivative, loosely speaking, tells us how much a multi-variable function will change when holding other variables constant. For instance, suppose we are given
$U \left(A , t\right) = {A}^{2} t$
Where $U$ is the utility (happiness) function of a particular product, $A$ is the amount of product, and $t$ is the time the product is used for.

Suppose the company which manufactures the product would like to know how much more utility they can get out of it if they increase the lifespan of the product by 1 unit. The partial derivative will tell the company this value.

The partial derivative is generally denoted by the lowercase Greek letter delta ($\partial$), but there are other notations. We will be using $\partial$ for now.

If we're trying to find how much the utility of the product changes with a 1 unit increase in time, we are computing the partial derivative of utility with respect to time:
$\frac{\partial U}{\partial t}$

To compute the PD, we hold other variables constant. In this case, we treat ${A}^{2}$, the other variable, as if it were a number. Recall from introductory calculus that the derivative of a constant times a variable is just the constant. It's the same idea here: the (partial) derivative of ${A}^{2}$, a constant, times $t$, the variable, is just the constant:
$\frac{\partial U}{\partial t} = {A}^{2}$

Thus, a 1 unit increase in the time the product is used produces ${A}^{2}$ more utility. In other words, the product becomes more satisfactory if it is able to be used more often.

There is much, much more to be said about partial derivatives - in fact, entire undergraduate and graduate courses can be devoted to solving just a few types of equations involving partial derivatives - but the basic idea is that the partial derivative tells us how much one variable changes when the other ones remain the same.