# 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. 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.