Calculo Diferencial

08/01/2010

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The differential

Andreas Koch

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What is a differential?
Exercises
Calculating a differential
Practical uses of differentials
Exercises

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08/01/2010

If y =f(x) is a differentiable function in an open interval that contains x, the differential of x (dx) its any
real number.
The differential of y (dy) is expressed as
Function

y = x2
2

y = x +1
y=x +1
2x −1

dy = f ' ( x )dx .
Derivative

Differential

dy
= 2x
dx
1
−
dy 1 2
= x + 1 2 (2 x )
dx 2

(

)

dy (2 x − 1) − ( x + 1)(2 )
=
dx
(2 x − 1)2

dy = 2 x dx
dy =dy =

x
x2 + 1

dx

−3
dx
(2 x − 1)2

In simple words a differential is a very small change.

y' =

dy
∆y
= lim
= f ' ( x)
dx ∆x →0 ∆x

dy = f ' ( x) dx
∆x → dx

So, inother words we can conclude that the differential of a function y=f(x) at one point is the increment
of the tangent line to the curve at that point.
Adapted from:http://www.bymath.com/studyguide/ana/sec/ana3.htm on 10/01/09.

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To find the differential of any function:
-Find the first derivative
-Multiply the derivative by dx
Get in pairs and find the differential of thefollowing functions:
1.
2.

3.

4.
5.

1.

Find the differential of the function

2.

Find the differential of the function

3.

Find the differential of the function

4.

Find thedifferential of the function

5.

Find the differential of the function

if x = 4 and dx = 0.2
if x = 6 and dx = 0.08
if x = 2.5 and dx = 0.5

if x = 8 and dx = 0.01
if x = 0.7 and dx =0.2

Homework:
1.1 - “The differentials”
Please do exercises 1 to 22, give random values to x and dx and solve.

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The differentials can also be used in approximations, geometryproblems and physics, let’s see:
1. “Linear Approximations and Differentials”

Zooming in . . .

A curve lies very close to its tangent
line near the point of tangency.

This observation is...