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mtanti
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I always wondered how you can differentiate a^x from first principles with the limit as dx approaches zero but I never managed to simplify it far enough to separate dx on a different term. Can anyone help?
HallsofIvy said:Of course, the hard part is showing that
[tex]\lim_{h\rightarrow 0}\frac{a^h-1}{h}= ln(a)...[/tex]
The simplest way to prove this rigourously is by introducing the ugly-looking function Exp(x):mtanti said:Hey but that's not fair, any up to standard student can get there on his/her own! How do you actually solve the hard part? I think it's interesting for all those starting calculus...
Tell me this at least... Can you diffentiate *every* equation from first principles? Even implicite ones?
Philip Wood said:If we differentiate log x (to any base) from first principles, after a few lines of algebra we find it to be (1/x) times the log of the limit as h approaches zero of (1 + h)^(1/h). But this limit is the familiar definition of e. Job done. Using this result its then easy to find (using chain rule or whatever) the derivative of the inverse function, i.e. the exponential function.
I'd say this counts as differentiating the exponential function from first principles. The limit Radou uses, is in my opinion, less well known that the limit I've referred to above.
"D/dx first principles" refers to the method of finding the derivative of a function using the limit definition of the derivative. This involves taking the limit as the change in the input variable approaches zero.
Understanding "D/dx first principles" allows us to find the derivative of any function, even if it is not a simple polynomial. This is a fundamental concept in calculus and is essential for solving complex problems in physics, engineering, and other fields.
To use "D/dx first principles" to find the derivative of a function, we first write out the limit definition of the derivative. Then, we plug in the function and simplify the expression as much as possible. Finally, we take the limit as the change in the input variable approaches zero to find the derivative.
One limitation of using "D/dx first principles" is that it can be a time-consuming process, especially for more complex functions. Additionally, it may not be possible to find the derivative of certain functions using this method, as the limit may not exist or may be difficult to evaluate.
"D/dx first principles" is the most basic method for finding derivatives and is the foundation for other methods, such as the power rule, product rule, and chain rule. It allows us to understand the concept of a derivative and serves as a starting point for more advanced techniques.