Derivative and integral of the exponential e^t

In summary, the question is whether the integral of e^t is the same as the derivative. The answer is that while the derivative always produces a unique function, the integral produces an equivalence class of functions. This is why "e" is defined as it is, as the derivative and integral of e^t are always the same.
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Is the integral of e to the t the same as the derivative?
If I take the exponential function e^t and take the derivative, I think I get the same e^t. Even if I keep doing it over and over, second, third derivative, etc. My admittedly naive question, though, is this symmetric? Meaning...if I take the the integral of e^t, do I just get the reverse or do I have t deal with an infinity of constants because it is an indefinite integral?
 
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DiracPool said:
Summary: Is the integral of e to the t the same as the derivative?

If I take the exponential function e^t and take the derivative, I think I get the same e^t. Even if I keep doing it over and over, second, third derivative, etc. My admittedly naive question, though, is this symmetric? Meaning...if I take the the integral of e^t, do I just get the reverse or do I have t deal with an infinity of constants because it is an indefinite integral?

Yes, an indefinite integral produces an equivalence class of functions, not a unique function.

This is important as you can see from the following example.

We want to find functions, ##f(t)## that satisfy ##f''(t) = e^t##. Clearly ##f(t) = e^t## is one such function. But, ##f(t) = e^t + At + B## is also a solution to this differential equation, where ##A, B## are any two constants.

Another way to look at this is to say that if you integrate ##f''(t) = e^t## you do not get a unique result.

Differentiation, on the other hand, always gives a unique function as a result.
 
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Yes, the derivative of [itex]e^x[/itex] is again [itex]e^x[/itex] and so the nth derivative, for all n, of [itex]e^x[/itex] is [itex]e^x[/itex] and all integrals of [itex]e^x[/itex] are again [itex]e^x[/itex]. In fact, that is why "e" is defined as it is. One can show that the derivative or [itex]a^x[/itex], for a any positive number, is [itex]C_aa^x[/itex] where "[itex]C_a[/itex]" is a constant (independent of x) that depends upon a. "e" is the unique number such that [itex]C_e= 1[/itex].
 

1. What is the derivative of e^t?

The derivative of e^t is e^t. This means that the derivative of the exponential function is itself.

2. How do you find the integral of e^t?

The integral of e^t is e^t + C, where C is a constant. This can be found using the power rule for integration, where the power of e is kept the same and the constant is added at the end.

3. What is the relationship between the derivative and the integral of e^t?

The derivative and integral of e^t are inverse operations of each other. This means that the integral of e^t is the reverse process of finding the derivative of e^t.

4. Can e^t be integrated using substitution?

Yes, e^t can be integrated using substitution. The substitution u = e^t can be used to simplify the integral and make it easier to solve.

5. How is the graph of e^t related to its derivative and integral?

The graph of e^t is a curve that increases rapidly as t increases. The derivative of e^t is also a curve, but it represents the rate of change of the original curve. The integral of e^t is the area under the curve, which can be used to find the total change over a specific interval.

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