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hackensack
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Hey, I've been tiring over this problem for a while...how do you get the derivitive of y=2^x using the definition of derivitive? Any help would be greatly appreciated.
-Hackensack-
-Hackensack-
Hmm... You mean from first principles?hackensack said:Hey, I've been tiring over this problem for a while...how do you get the derivitive of y=2^x using the definition of derivitive? Any help would be greatly appreciated.
-Hackensack-
hackensack said:Hey, I've been tiring over this problem for a while...how do you get the derivitive of y=2^x using the definition of derivitive? Any help would be greatly appreciated.
-Hackensack-
cookiemonster said:I imagine that using a series expansion would involve the derivative anyway. May as well use L'Hospital's if you can do that...
cookiemonster
Gokul43201 said:I haven't tried this but i imagine this {lim(2^h - 1)/h = ln(2), as h -->0} can also be proven inductively.
HallsofIvy said:If you were referring to wisky40's work, that was just an expansion using the binomial theorem, known long before calculus itself (and regularly used to find derivatives in the early days of calculus).
cookiemonster said:The binomial expansion is simply the taylor series of (x + 1)^n, is it not?
cookiemonster
Zurtex said:[tex]\frac{dy}{dx} \left( \frac{1}{y} \right) = \ln (a)[/tex]
[tex]\frac{dy}{dx} = y \ln(a)[/tex]
The formula for finding the derivative of 2^x is d/dx(2^x) = ln(2) * 2^x. This means that the derivative of 2^x is equal to the natural logarithm of 2 (ln(2)) multiplied by 2^x.
The derivative of 2^x represents the rate of change of the function 2^x at a specific point. It tells us how much the function is changing at that point and in what direction (increasing or decreasing).
The natural logarithm of 2 (ln(2)) is significant because it is the constant multiplier in the derivative formula for 2^x. This value ensures that the derivative of 2^x is always proportional to the original function, making it a useful tool in many applications.
Yes, the derivative of 2^x can be negative. This means that the function 2^x is decreasing at that specific point. However, the derivative can also be positive or zero, depending on the value of x and the behavior of the function.
The derivative of 2^x is used in many real-world applications, particularly in the fields of economics, physics, and engineering. It can be used to model exponential growth and decay, calculate rates of change in various processes, and optimize systems for maximum efficiency.