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AMF8
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why does d e^x / dx = e^x ?
AMF8 said:why does d e^x / dx = e^x ?
benorin said:Is power series too much?
0 = e^x ?Integral said:As Binorin said.
[tex] e = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} +\frac {x^4} {4!} + ... [/tex]
Now differentiate.
Robokapp said:Power rule doesn't work for variables.
e^x=> xe^(x-1) is not right as far as i can tell.
Hehe, its probably just a typo , it should be:neutrino said:0 = e^x ?
You may be correctIgor_S said:Hehe, its probably just a typo , it should be:
[tex]e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots[/tex]
(offcourse :)
Orion1 said:Can it not be argued that that the solution is derived from theorem?
Of course, without the knowledge that dex/dx=ex, you wouldn't know that the anti-derivative of 1/x is ln(x)!VikingF said:[tex] y' = y[/tex]
[tex]\frac{dy}{dx}=y[/tex]
We multiply with [tex] dx [/tex] on each side and divide by [tex] y [/tex]:
[tex] \frac{dy}{y} = dx [/tex]
Integrals please!
[tex] \int{\frac{dy}{y}} = \int{dx} [/tex]
[tex]lny = x+C'[/tex]
Out of knowledge of the natural logarithm:
[tex]y = e^{x+C'}[/tex]
[tex]y = e^xe^{C'}[/tex]
[tex]C=e^{C'}[/tex]
[tex]y = Ce^x[/tex]
That's it!
apmcavoy said:Of course, without the knowledge that dex/dx=ex, you wouldn't know that the anti-derivative of 1/x is ln(x)!
cepheid said:It looks circular, but you could go the other way 'round the circle i.e. start by defining a function as a solution to the integral:
[tex] f(x) = \int{\frac{dx}{x}} [/tex]
Without evaluating the integral (because you don't know how), you can figure out some of the properties of the function and see that it has all the properties of a logarithm function (is there some way to make that more rigourous?) Call it the natural logarithm (it arose naturally in our investigation of that integral). Then ask, what is the base of this mysterious logarithm? What are the properities of the inverse exponential function? Once you find out that this base crops up everywhere in the natural sciences, it makes sense that it is natural exponential function.
My first year calculus textbook did it both ways. Personally I like starting out with the exponential function (DH's post) better.
The derivative of a function is a measure of its instantaneous rate of change. In the case of e^x, its derivative is equal to itself, meaning that the rate of change of e^x at any point is equal to the value of e^x at that point. This can be mathematically proven using the definition of the derivative and the properties of exponential functions.
The derivative of e^x does not change because it is a constant value. This is due to the fact that e^x is a special type of exponential function called a natural exponential function, and its derivative is always equal to itself. This is one of the many properties of exponential functions that make them useful in many areas of science and mathematics.
The value of e, approximately equal to 2.718, is a fundamental constant in mathematics and is the base of natural logarithms. In the case of e^x, the derivative is equal to e^x because e is the base of the function. This means that the natural exponential function has a special property where its derivative is equal to itself, making it a useful function in many scientific and mathematical applications.
The derivative of e^x can be extended to other exponential functions by using the chain rule of differentiation. This means that the derivative of e^x can be applied to any function raised to the power of x, where x is a variable. This generalizes the derivative of e^x and allows us to find the derivative of other exponential functions.
Yes, the derivative of e^x has many real-life applications in fields such as physics, economics, and engineering. For example, it can be used to model population growth, radioactive decay, and compound interest. It is also used in differential equations, which are used to model many natural phenomena. Understanding the properties of the derivative of e^x is essential for solving these real-world problems.