d/dx e^x
The expression “d/dx e^x” represents the derivative of the function e^x with respect to x
The expression “d/dx e^x” represents the derivative of the function e^x with respect to x. To find this derivative, we can use the basic derivative rule for exponential functions.
The derivative of e^x with respect to x is simply e^x itself. In other words:
d/dx e^x = e^x
This means that the rate of change of the function e^x with respect to x at any point is equal to the value of the function itself at that point.
To give a more detailed explanation, let’s use the definition of the derivative.
We start by using the definition of a derivative:
d/dx f(x) = lim(h->0) [f(x+h) – f(x)] / h
Applying this definition to our specific function, f(x) = e^x, we have:
d/dx e^x = lim(h->0) [e^(x+h) – e^x] / h
Now, let’s simplify this expression:
d/dx e^x = lim(h->0) [e^x * e^h – e^x] / h
Next, we can factor out e^x:
d/dx e^x = e^x * lim(h->0) [e^h – 1] / h
The term inside the limit is a standard limit that evaluates to 1 as h approaches 0:
lim(h->0) [e^h – 1] / h = 1
Finally, substituting this value back into the original expression, we get:
d/dx e^x = e^x * 1 = e^x
Therefore, the derivative of e^x with respect to x is e^x.
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