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

Recall the definition of the derivative

The derivative of a function is defined as follows: $$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

Apply the definition to the given function

Now, let's apply the definition of the derivative to the given function \(f(x) = e^{-x}\): $$ f'(x) = \lim_{h \to 0} \frac{e^{-(x+h)} - e^{-x}}{h} $$

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

Now factor out the \(e^{-x}\) term from the numerator: $$ f'(x) = \lim_{h \to 0} \frac{e^{-x}(e^{-h} - 1)}{h} $$ $$ f'(x) = e^{-x} \cdot \lim_{h \to 0} \frac{e^{-h} - 1}{h} $$ #b. Proving the limit is equal to -1#

Use given hint to compute the limit

It is given that \(\lim_{h \to 0} \frac{e^{h} - 1}{h} = 1\). We need to find the limit of another expression: $$\lim_{h \to 0} \frac{e^{-h} - 1}{h}$$ Let \(u = -h\). Then, as \(h \to 0\), \(u \to 0\). So, we have: $$ \lim_{u \to 0} \frac{e^{u} - 1}{-u} = -1 \cdot \lim_{u \to 0} \frac{e^{u} - 1}{u} = -1 \cdot 1 = -1$$ #c. Finding the derivative of \(f(x) = e^{-x}\) using parts (a) and (b)#

Substitute the limit value in part (a)

Now, substitute the value of the limit we found in part (b) into the expression from part (a): $$ f'(x) = e^{-x} \cdot (-1) $$

Write the final derivative

Therefore, the derivative of the function \(f(x) = e^{-x}\) is: $$ f'(x) = -e^{-x} $$

Key concepts

Definition of the Derivative

The derivative of a function is a fundamental concept in calculus. It measures how a function changes as its input changes. Imagine slipping on a shoe. The derivative gives us the rate at which the shoe fits more or less snugly as the size changes. In mathematical terms, the derivative of a function \( f(x) \) at a point \( x \) is a limit:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

The expression \( f(x+h) - f(x) \) calculates the change in the function's value. The \( h \) part is a small change in the \( x \) value. Taking the limit as \( h \) approaches zero gives us the precise rate of change, or the derivative. This formula helps us see how a tiny shift in \( x \) affects \( f(x) \). When you apply this to an exponential function like \( f(x) = e^{-x} \), you enter its powerful world of calculus where even tiny changes have clear, computable effects.

Limit of a Function

Limits help us understand what happens to a function as it approaches a particular point or value. These are incredibly useful in calculus for calculating derivatives and integrals. Essentially, a limit evaluates the behavior of a function as its inputs get infinitely close to some point. To find such a limit for a derivative, you'd look at an expression like this:

• \( \lim_{h \to 0} \frac{e^{-h} - 1}{h} \)

In our exercise, we used a handy hint: \( \lim_{h \to 0} \frac{e^{h} - 1}{h} = 1 \). Substituting back, with a change of variables, shows that the original limit equals \(-1\). This helps in confirming parts of the derivative calculation and helps us visualize the function's behavior near that point. Understanding limits is crucial because they form the backbone of calculating derivatives and integrals which describe rates of change.

Exponential Function Continuity

Exponential functions, such as \( e^x \), are continuous. Continuity means there are no breaks, jumps, or holes in the graph of the function. For a function to be continuous at a point \( x \), three things must be true:

• \( f(x) \) is defined at the point.
• \( \lim_{y \to x} f(y) \) exists, meaning it approaches a particular value as \( y \) gets close to \( x \).
• \( \lim_{y \to x} f(y) = f(x) \), linking the value approached by the limit to the actual function value.

Exponential functions maintain this continuity over their entire domain. This property is essential when calculating limits and derivatives smoothly. Knowing that \( e^x \) is continuous allows us to substitute limits directly, ensuring correct derivative calculation without unexpected jumps or oddities. Embracing continuity simplifies analysis, especially with calculus operations.