Question

Computing the derivative of \(f(x)=e^{-x}\) a. Use the definition of the derivative to show that $$\frac{d}{d x}\left(e^{-x}\right)=e^{-x} \cdot \lim _{h \rightarrow 0} \frac{e^{-h}-1}{h}$$ b. Show that the limit in part (a) is equal to \(-1 .\) (Hint: Use the facts that \(\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}=1\) and \(e^{x}\) is continuous for all \(x\).) c. Use parts (a) and (b) to find the derivative of \(f(x)=e^{-x}\)

Short answer

Answer: The derivative of the function \(f(x) = e^{-x}\) is: \(f'(x) = -e^{-x}\).

Step-by-step solution

Use the definition of the derivative

The definition of the derivative of a function \(f(x)\) is: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ Our function is \(f(x) = e^{-x}\). We must compute \(f(x+h)\) and insert it into the formula: $$f(x + h) = e^{-(x + h)} = e^{-x}e^{-h}$$ Now, substitute the function expressions into the derivative definition: $$f'(x) = \lim_{h \to 0} \frac{e^{-x}e^{-h} - e^{-x}}{h}$$

Factor out \(e^{-x}\)

In the limit expression, factor out the common factor \(e^{-x}\): $$f'(x) = e^{-x} \lim_{h \to 0} \frac{e^{-h} - 1}{h}$$

Evaluate the limit

We are given a hint to use the fact that: $$\lim_{h \to 0} \frac{e^{h} - 1}{h} = 1$$ Now recognize that the given limit can be rewritten using this fact by noting that \(e^{-h} = \frac{1}{e^h}\): $$\lim_{h \to 0} \frac{e^{-h} - 1}{h} = \lim_{h \to 0} \frac{\frac{1}{e^h} - 1}{h} = \lim_{h \to 0} \frac{1 - e^h}{h\cdot e^h}$$ Multiply both the numerator and denominator by \(-1\) so we can use the given fact: $$\lim_{h \to 0} \frac{1 - e^h}{h\cdot e^h} = \lim_{h \to 0} \frac{e^h - 1}{ (-h)\cdot e^h}$$ Now, use the fact that the limit of products is the product of limits and that \(e^x\) is continuous for all \(x\): $$\lim_{h \to 0} \frac{e^h - 1}{ (-h)\cdot e^h} = -\lim_{h \to 0} \frac{e^h - 1}{h} \cdot \lim_{h \to 0} e^h = -1 \cdot 1 = -1$$

Find the derivative

Now, substitute the value of the limit back into the expression for \(f'(x)\) from Step 2: $$f'(x) = e^{-x} \cdot (-1)$$ So the derivative of the function \(f(x) = e^{-x}\) is: $$f'(x) = -e^{-x}$$

Key concepts

Definition of the Derivative

The definition of the derivative is a cornerstone concept in calculus that measures how a function changes as its input changes. It's represented as the limit of the difference quotient as the increment approaches zero. This can be formally written as:

$$f'(x) = \frac{d}{d x}[f(x)] = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
This expression calculates the instant rate of change at a particular point, which is also the slope of the tangent line to the function at that point. In the context of the exponential function, we see this definition applied to compute the derivative of functions like \(e^{-x}\), which leads to exploring limits as well.

Limits in Calculus

Limits are fundamental to calculus and function to describe the behavior of a function as it approaches a particular point. For instance, when we're looking at the derivative of an exponential function, limits help us to find out what happens as a variable gets infinitesimally small, which is crucial for understanding the nature of the function's rate of change.

In our example, the limit \(\lim_{h \to 0} \frac{e^{-h} - 1}{h}\) appears in the process of differentiating \(e^{-x}\). By understanding and manipulating limits, we can navigate complex functions and find derivatives, often using known limits like \(\lim_{h \to 0} \frac{e^{h} - 1}{h} = 1\) to simplify our work.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. Mathematically, this is represented by functions like \(f(x) = e^{-x}\), where \(e\) is the base of the natural logarithm.

Understanding the derivative of such functions is crucial because it allows us to model dynamic processes, like radioactive decay or cooling temperatures. The derivative \(-e^{-x}\) shows us that the rate of decay is also an exponential function, and this negative sign indicates that the quantity is decreasing over time.

Continuity of Exponential Functions

Continuity of a function at a point means that there is no abrupt change in its value – it’s smooth without any breaks, holes, or jumps. Exponential functions, like \(e^x\), are continuous over their entire domain, which contributes to their predictability and the ease of calculating their derivatives.

This continuity property comes into play when solving limits involving exponential functions and assures that we can manipulate the function's expression when approaching limits, as we do when evaluating the derivative of the exponential decay function.