How is the dx quantity derived in using differentials to approximate values?

[JoshHolloway]

This is a problem from a Calc 1 textbook, I just can't figure out where they get dx from. The question is:
Use differentials to approximate the value of the expresesion.
\sqrt{99.4}

the answer in the solution manual says:
Let f(x)=\sqrt{x}, x=100, dx=-.6

then it solves the problem. But what I don't understand is where the heck is this dx quantity derived from?

 

[HallsofIvy]

f(x)=\sqrt{x}, x=100, dx=-.6

Your textbook is using the fact that df= f'(x)dx so that f(x+dx) is approximately
f(x)+ df= f'(x)dx.
Here, you want to evaluate f(99.4) and it easy to see that f(100)= 10 so take x= 100 and x+ dx= 99.4. What is dx?

 

[JoshHolloway]

Wow, I get it. It's so simple. Dx is -0.6. This also makes it more clear why it is called a differential. Thanks Ivy! Hey Ivy, are these differentials the same thing that differential equations are based on?

 

[mustafa]

Wow, I get it. It's so simple. Dx is -0.6. This also makes it more clear why it is called a differential. Thanks Ivy! Hey Ivy, are these differentials the same thing that differential equations are based on?

Though similar, they are not exactly the same thing.

In terms of differentials df=f'(x)dx is true but taking dx= -0.6 is only an approximation. In the true sense a differential can be considered as an infinitesimally small change in the variable.

You might be knowing the Taylor's series expansion of f(x+h)

f(x+h) = f(x) + hf'(x) + (h^2)/2! f"(x) + (h^3)/3! f"'(x) + ...

If h is very small relative to x, the first order approximation can be obtained as f(x+h) = f(x) + hf'(x), which is the same as the equation with differentials
f(x+dx) = f(x) + f'(x)dx

What I want to imply is that the dx in your question is an approximation but differentials in a differential equation do not signify any approximations.

 

[HallsofIvy]

Strictly speaking, differentials, like "dx", are "infinitesmals". When you write something like "dx= -0.6", it really is \Delta x, meaning a small change in x. One way of defining the derivative is lim_{\Delta x->0} \frac{\Delta y}{\Delta x}. You can then use \Delta x to approximate dx:
\frac{\Delta y}{\Delta x} is approximately \frac{dy}{dx} so \Delta y is approximately \(\frac{dy}{dx}\)\Delta x. The smaller \Delta x is, the better the approximation.