Differentials

Suppose that we are given function `y=f(x)`. Consider interval `[a,a+Delta x]`. Corresponding change in `y` is `Delta y=f(a+Delta x)-f(a)`.

We are interested in the following question: is there exist constant `C` such that `Delta y~~C Delta x` when `Delta x->0`?

Let's go through a couple of examples.

Example 1 . Area of circle with radius `r` is `A=pir^2`. If we increase `r` by `Delta r` then corresponding increase in `A` is `Delta A=pi(r+Delta r)^2-pir^2=2pirDelta r+(Delta r)^2`. When `Delta r->0` we have that `(Delta r)^2` becomes very small, so we can write that `Delta A~~2pirDelta r`.

Example 2. Volume of sphere with radius `r` is `V=4/3pir^3`. If we increase `r` by `Delta r` then corresponding increase in `V` is `Delta V=4/3pi(r+Delta r)^3-4/3pir^3=4pir^2Delta r+4pir(Delta r)^2+4/3pi(Delta r)^3`. When `Delta r->0` we have that both `(Delta r)^2` and `(Delta r)^3` become very small, so we can write that `Delta V~~4pir^2Delta r`.

In above examples we can see the same thing. In Example 1 when `A=pir^2` we have that `Delta A~~2pirDelta r`, but `2pir` is derivative of `pir^2` with respect to `r`. In Example 2 when `V=4/3pir^3` we have that `Delta V~~4pir^2Delta r`, but again `4pir^2` is derivative of `4/3 pir^3` with respect to `r`.

In fact this is true for general function.

Definition. Function `y=f(x)` is called differentiable at point `a` if and only if it has finite derivative `f'(a)`.

Fact. If function `y=f(x)` is differentiable then `Delta y~~f'(x)Delta x` or `dy~~Delta y`. Expression `f'(x)Delta x` is called differential of the function `f(x)` and denoted by `dy`. So, `dy=f'(x)Delta x`.

Differential of the independent variable `x` is just its increase, i.e. `dx=Delta x`.

Thus, differential of the function can be rewritten as `dy=f'(x)dx` or `f'(x)=(dy)/(dx)`.

Now we can treat `(dy)/(dx)` as ratio, not as just symbol.differential

Now, let's see geometric meaning of differential.

Let `P=(x,f(x))` and `Q=(x+Delta x,f(x+Delta x))` be points on the graph `f` and let `Delta x=dx` . The corresponding change in `y` is `Delta y=f(x+Delta x)-f(x)` .

The slope of the tangent line `PA` is the derivative `f'(x)`. Thus, the directed distance from A to B is `f'(x)dx=dy`. Therefore, `dy` represents the amount that the tangent line rises or falls (the change in the linearization), whereas `Delta y` represents the amount that curve `y=f(x)` rises or falls when `x` changes by an amount `dx`. Notice that the approximation `Delta y~~dy` becomes better as `Delta x` becomes smaller.

Since `Delta y=f(a+Delta x)-f(a)` then now we can write that `f(a+dx)-f(a)~~dy`.

This can be rewritten as `f(a+dx)~~f(a)+dy`.

This formula is used for approximating values of functions.

Example 3. Approximate value of `sqrt(16.05)` using function `f(x)=sqrt(x+4)` .

`dy=f'(x)dx=1/(2sqrt(x+4))dx`

If `a=12` and `dx=Delta x=0.05` then

`dy=1/(2sqrt(12+4))0.05=0.00625`.

Now, `sqrt(16.05)=f(12.05)=f(12+0.05)~~f(12)+dy=4+0.00625=4.00625`.