Refer to the following: Recall that the derivative of f can...

Refer to the following: Recall that the derivative of $f$ can be found by letting $h \rightarrow 0$ in the difference quotient $\frac{f(x+h)-f(x)}{h} .$ In calculus we prove that $\frac{e^{h}-1}{h}=1,$ when $h$ approaches $0 ;$ that is, for really small values of $h, \frac{e^{h}-1}{h}$ gets very close to 1. Use this information to find the derivative of $f(x)=e^{x}$

Refer to the following:
Recall that the derivative of $f$ can be found by letting $h \rightarrow 0$ in the difference quotient $\frac{f(x+h)-f(x)}{h} .$ In calculus we prove that $\frac{e^{h}-1}{h}=1,$ when $h$ approaches $0 ;$ that is, for really small values of $h, \frac{e^{h}-1}{h}$ gets very close to 1.
Use this information to find the derivative of $f(x)=e^{x}$

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Key Concepts

Derivative

The derivative of a function is a measure of its instantaneous rate of change and is defined as the limit of the difference quotient as the increment approaches zero. This concept is central to calculus and allows us to understand how functions behave locally.

Difference Quotient

The difference quotient, given by (f(x+h) - f(x)) / h, is used to compute the average rate of change of a function over an interval. By taking the limit as h approaches zero, we obtain the derivative, which represents the instantaneous rate of change.

Limits

Limits are a fundamental concept in calculus used to analyze the behavior of functions as the input approaches a particular value. They are essential for defining the derivative and for dealing with small increments in the difference quotient.

Exponential Function

The exponential function, particularly e^x, has unique properties in calculus. One of its key properties is that the rate of change of e^x is equal to itself. This is demonstrated using the limit definition of the derivative, where the limit of (e^h - 1)/h as h approaches zero equals 1, leading to the result that d/dx e^x = e^x.

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