# Derivative of e

1. Nov 23, 2004

### UrbanXrisis

the derivative of e^u = e^u du/dx

so the derivative of e would be 0 since:
e^1 * 0 = 0

is this correct?

well, I knot it's not correct because derivative of e = e. It's because 1 is a constant right?

derivative of e^(1+x) = e^(1+x) * 2

is that correct?

Last edited: Nov 23, 2004
2. Nov 23, 2004

### heavyarms

The derivative of e is 0 since e by itself is a constant (just like pi).

The derivative of e^(1+x) is [e^(1+x)][1][dx].

The derivative of e^x is (e^x)(dx)

Hope that clarifies things up for you.

3. Nov 23, 2004

### UrbanXrisis

"The derivative of e^(1+x) is [e^(1+x)][1][dx]."

wouldn't it be The derivative of e^(1+x) is [e^(1+x)][1]+[dx].

4. Nov 23, 2004

### vincebs

The derivative must always be taken with respect to some variable. You can't just say "take the derivative of e^u".

Using u as the variable, the derivative df(u)/du of the function f(u) = e^u is e^u.
Therefore the derivative of e^u with respect to u AT u = 1 is df(1)/du = e^1 = e.

If u is actually a function itself, say u = g(x), then when we take the derivative of e^u, we are taking it with respect to x. Then e^u = e^(g(x)) = f(x)

Now the derivative according to the chain rule is df(x)/dx = e^(g(x)).dg(x)/dx = e^u . du/dx. When you said that e^u du/dx = 0, this is only true if u is constant no matter what x is.

You can see here that if f = e^u, df/du = e^u but df/dx = e^u du/dx, two different things.

5. Nov 23, 2004

### sal

Erk. Let's try to keep straight what we're doing here.

The "derivative", all by itself, doesn't mean anything -- you need to say what you're differentiating with respect to.

The derivative of the number "e" with respect to x is zero because e isn't a function of x (or anything else -- it's a constant).

The derivative of the function ex with respect to x is the function ex, or more clearly,

$$\frac{d(e^x)}{dx}|_u = e^u$$

In other words, the the derivative of ex with respect to x, evaluated at x=u, is the value eu.

But please get those "dx" terms out of the derivative -- they have no business there. In basic differential calculus you should treat the "dx" as a reminder of what you're differentiating with respect to, and a nice reminder that it's a ratio of small values that you're talking about, but you shouldn't have it appearing by itself.

$$\frac{d(e^{1+x})}{dx} = e^{1+x}$$
$$\frac{d(1+x)}{dx} = 1$$