the number e is by definition the number which raised to x will produce a graph such that at every point on the graph the slope of the graph is equal to x.
In otherwords, the answer to your question is because that is the way e is defined.
IDK... I saw e^x defined as a power series.
Consider: Let y = e^x
Since x is real (by assumption) then
ln(y) = x, where ln(t) is the natural logarithm of t. Now differentiate with respect to x
1/y * (dy/dx) = 1 Multiplying both sides by y, we get
dy/dx = y = e^x. This fits with the first definition but is more rigorous.
The first derivative of e to the x power is e to the power of x.
Well the number e, raised to 6 (e^6) is just a number (a constant), so you integrate a constant times dx gives you that constant times x + C --> x*e^6 + C
The derivative of ex is ex
x e^x +C
That's because powers that involve the power "e", and logarithms to the base "e", are simpler than other powers or logarithms. For example: the derivative of ex is ex, while a derivative with other bases is more complicated; while the derivative of the natural logarithm (ln x, or logex) is 1/x.
The first derivative of e to the x power is e to the power of x.
Euler's constant, e, has some basic rules when used in conjunction with logs. e raised to x?æln(y),?æby rule is equal to (e raised to ln(y) raised to x). e raised to ln (y) is equal to just y. Thus it becomes equal to y when x = 1 or 0.
e^(-2x) * -2 The derivative of e^F(x) is e^F(x) times the derivative of F(x)
It is possible.
2.71828183 ==So the derivative of a constant is zero.If you have e^x, the derivative is e^x.
Well the number e, raised to 6 (e^6) is just a number (a constant), so you integrate a constant times dx gives you that constant times x + C --> x*e^6 + C
- e^- X
The derivative of ex is ex
e^[ln(x^2)]=x^2, so your question is really, "What is the derivative of x^2," to which the answer is 2x.
y = e^ln x using the fact that e to the ln x is just x, and the derivative of x is 1: y = x y' = 1
I suppose you mean of e-x? It is -e-x + C.
d/dx (e-x) = -e-x