Problem 121 - Precalculus

Question

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 \(f(x)=e^{x}\).

Step-by-Step Solution

Answer

The derivative of \(f(x) = e^x\) is \(e^x\).

Step 1: Set Up the Difference Quotient

The derivative of the function \(f(x) = e^x\) is defined by the limit of the difference quotient as \(h\) approaches 0. So, set up the expression: \[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} = \lim_{{h \to 0}} \frac{e^{x+h} - e^x}{h}. \]

Step 2: Simplify the Exponential expression

Use the property of exponents to simplify \(e^{x+h}\). Rewriting this gives: \[ \lim_{{h \to 0}} \frac{e^x \cdot e^h - e^x}{h}. \]

Step 3: Factor Out Common Terms

Notice that \(e^x\) is a common factor in the numerator. Factor it out: \[ \lim_{{h \to 0}} \frac{e^x(e^h - 1)}{h}. \]

Step 4: Evaluate the Limit Using a Given Result

Apply the fact that \( \frac{e^h - 1}{h} \to 1 \) as \( h \to 0 \). Substitute this in: \[ f'(x) = e^x \cdot \lim_{{h \to 0}} \frac{e^h - 1}{h} = e^x \cdot 1. \]

Step 5: Finalize the Derivative

The limit simplifies and the derivative of \(f(x) = e^x\) is \[ f'(x) = e^x. \]

Key Concepts

Difference Quotient

The concept of the difference quotient is central to understanding how derivatives work in calculus. The difference quotient is expressed as \( \frac{f(x+h) - f(x)}{h} \). It gives the average rate of change of the function \( f \) over the interval from \( x \) to \( x + h \).
As \( h \) gets closer to zero, the difference quotient approaches the instantaneous rate of change of the function at point \( x \), which is the derivative of the function. To put it simply, using the difference quotient allows us to see how a function's output changes minutely when there's a tiny change in input.

If we return to our example with \( f(x) = e^x \), the difference quotient helps in setting up the limit expression \( \lim_{{h \to 0}} \frac{e^{x+h} - e^x}{h} \). This expression is the foundation for finding the derivative of the function. Understanding difference quotients makes grasping derivatives much easier.

Exponential Function

The exponential function, particularly \( f(x) = e^x \), is one of the fundamental functions in calculus. It has unique properties, one of them being that its derivative is the same as the function itself: \( f'(x) = e^x \).
The constant \( e \) is an irrational number approximately equal to 2.71828, and it is the base of the natural logarithms. This base is particularly important because of the convenient property that \( e^{h} \) approaches 1 as \( h \) approaches 0, leading to simplifications when calculating derivatives.

In many contexts, especially involving growth and decay, the exponential function is applicable, making the understanding of its differentiation very useful. In the exercise solution, we utilized the property of exponents coupling it with the rule for limits to find that the derivative of \( e^x \) simplifies easily to \( e^x \). The exponential function's simplicity in differentiation is a significant attribute that makes it so widely used in mathematical analysis.

Limit

The limit is an essential concept that helps us find the value that a function approaches as its input approaches a certain point. In calculus, limits are used to define both derivatives and integrals.
For instance, in our exercise, we used the limit \( \lim_{{h \to 0}} \frac{e^h - 1}{h} = 1 \) to solve for the derivative of the exponential function \( e^x \). This particular limit shows how a very small change in the power of \( e \) results in the expression \( \frac{e^h - 1}{h} \) reducing to the value 1 as \( h \) gets very close to zero.

This property makes the calculation straightforward and illustrates the power of limits in handling expressions that seem undefined at first. Limits help to bridge the gap between values that can be calculated directly and those that require more theoretical understanding. They are a toolkit for making sense of rates of change and patterns in calculus, particularly when expressions become tricky to evaluate directly.