Because the derivative of e^x is e^x (the original function back again). This is the only function that has this behavior.
using Laplace transform, we have: sY(s) = Y(s) + 1/(s2) ---> (s-1)Y(s) = 1/(s2), and Y(s) = 1/[(s2)(s-1)]From the Laplace table, this is ex - x -1, which satisfies the original differential eq.derivative of [ex - x -1] = ex -1; so, ex - 1 = ex - x - 1 + xto account for initial conditions, we need to multiply the ex term by a constant CSo y = C*ex - x - 1, and y' = C*ex - 1, with the constant C, to be determined from the initial conditions.
I suppose you mean ex. The derivate is also ex.
e2x + ex = 0e2x = -exe2x / ex = -1ex = -1x = ln(-1)Actually, that's just a restatement of the original problem, but that'sas far as you can go with it, because there's no such thing as the log(or the ln) of a negative number. Your equation has no solution.
I'm not sure if you are asking about: e-ax2 or e-ax^2. Use the carets=^ in the future to raise something to a power.For e-ax2:You need to remember that: (ex)`= exalso: (eax)`= aeax, assuming a is a constant and not a function of x.Justrecognizeany constant drops down to the front.Therefore: (e-ax2)`=-2ae-ax2, treat the negative just as a constant or (-1)For e-ax^2: this one is a little bit trickier:We can just use u-substitution.Let u = x2.u` = 2xIf we substitute the u for x^2, we have: e-au , and we know the deriviate of that is -ae-2au, but we need to multiply by u` as well to get:(-ae-2au)(2x) = -2xae-2au, then we can just plug u back in.u = x2, so:-2xae-2au = -2xae-2ax^2And that is the answer.(e-ax^2)`= -2xae-2ax^2
I assume you mean ex ? If so, by definition: ∫ex dx = ex + C Most calculus textbooks have a table of integrals which will list the integrals of other common forms of exponential & logarithmic functions.
A number without a variable. Ex.) 5 is a constant, 5x is not.
In Calculus, differentiation is when you apply the theorems to get the derived equation at a given rate, for example you have the velocity function and if you take its derivative, it will give you an acceleration function related to its velocity. Derivatives are often denoted as f'(x) or y'. Integration on the other hand is undoing differentiation. for ex, if you integrate acceleration equation, it will give you a velocity equation.
∫ ex dx = ex + CC is the constant of integration.
The y-intercept. Ex: AX-BY=C The 'C' stands for constant and is the y-intercept. Was this helpful?
Because the derivative of e^x is e^x (the original function back again). This is the only function that has this behavior.
Differential Calculus is to take the derivative of the function. It is important as it can be applied and supports other branches of science. For ex, If you have a velocity function, you can get its acceleration function by taking its derivative, same relationship as well with area and volume formulas.
Keep in mind that the integral of ex = ex and 2x = ex / ln 2. You can then rewrite the exponent as u = 2x, convert dx to du, and from there it's pretty straightforward. (I've left off the "+ C" part, because you should just know that.)
a constant ex: Pi. it will always be 3.14159... it will never change in value.
using Laplace transform, we have: sY(s) = Y(s) + 1/(s2) ---> (s-1)Y(s) = 1/(s2), and Y(s) = 1/[(s2)(s-1)]From the Laplace table, this is ex - x -1, which satisfies the original differential eq.derivative of [ex - x -1] = ex -1; so, ex - 1 = ex - x - 1 + xto account for initial conditions, we need to multiply the ex term by a constant CSo y = C*ex - x - 1, and y' = C*ex - 1, with the constant C, to be determined from the initial conditions.
Constant means continuing non-stop.to remain in the same place without any movement
Has only one variable and one constant and is perpendicular to that variable's axis. Ex) y = 3