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**Leibniz notation**

Lately I've been having trouble with derivatives, specifically in the context of applying them to an ant problem (see this thread if you'd like to witness my slow decline). I have taken high school and college calc. I've learned many ways to perform the "hows" of calculus (how to find a derivative, how to find an integral, etc). But I admit it! I finally admit it to myself. I cannot

**apply**calculus to complex problems.

Could someone briefly explain Leibnitz notation and how you can use it to better understand the

**meaning**of derivative formulas? I already know how to take derivatives, so this is not really a concern. But when I first learned calculus they didn't stress the meaning(s) of notation enough, and now I feel left behind because I basically think of derivative notation as "f(x) with a prime symbol, like f'(x)."

I'll start slow. Can I define

(1) [tex]f(x) = x^2+3[/tex]

then say that equations two and three below mean exactly the same thing?

(2) [tex]f'(x) = 2x[/tex]

(3) [tex]\frac{d\ f(x)}{dx} = 2x[/tex]

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