Question

Find \(f^{\prime}(x)\) for each function. $$ f(x)=\frac{e^{-2}}{x} $$

Short answer

\(f^{\prime}(x) = \frac{-e^{-2}}{x^{2}}\)

Step-by-step solution

Rewrite the Function

The function is given as \( f(x) = \frac{e^{-2}}{x} \). This can be rewritten using the property of exponents as \( f(x) = e^{-2} \cdot x^{-1} \). This will make differentiation easier.

Identify the Differentiation Rules

For the function \( e^{-2} \cdot x^{-1} \), we will use the constant multiple rule and the power rule for differentiation. The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. The power rule states that \( \frac{d}{dx} [x^n] = nx^{n-1} \).

Apply the Constant Multiple Rule

Since \( e^{-2} \) is a constant, we apply the constant multiple rule first:\( \frac{d}{dx} [e^{-2} \cdot x^{-1}] = e^{-2} \cdot \frac{d}{dx} [x^{-1}] \).

Differentiate Using the Power Rule

Apply the power rule to \( x^{-1} \). By the power rule, \( \frac{d}{dx} [x^{-1}] = -1 \cdot x^{-2} = -x^{-2} \).

Combine the Results

Substitute the result of Step 4 into Step 3:\( f^{\prime}(x) = e^{-2} \cdot (-x^{-2}) = -e^{-2} \cdot x^{-2} \).

Simplify the Expression

Express the derivative in standard form:\( f^{\prime}(x) = \frac{-e^{-2}}{x^{2}} \). This is the simplified form of the derivative.

Key concepts

Constant Multiple Rule

When differentiating functions, the constant multiple rule is a very useful tool. It states that if you have a constant multiplied by a function, you can take the constant outside of the derivative. This means you differentiate the function part as usual and simply multiply your result by the constant afterwards. In our example, the function was rewritten as \( e^{-2} \cdot x^{-1} \). Here, \( e^{-2} \) is a constant, which makes it a perfect candidate for this rule. Thus, differentiating \( e^{-2} \cdot x^{-1} \) becomes simpler by first focusing on \( x^{-1} \) and keeping \( e^{-2} \) outside the differentiation process. This is like separating a tasty treat, working on one ingredient at a time, and then bringing everything together at the end for the final dish.

Power Rule

The power rule is a fundamental principle in differentiation that simplifies the process of finding derivatives for functions of the form \( x^n \). It tells us that the derivative of \( x^n \) is \( n \cdot x^{n-1} \). This transformation is due to bringing down the power as a coefficient and reducing it by one. Let's say you have \( x^{-1} \); applying the power rule means you multiply by \(-1\), the existing exponent, and then reduce the exponent by one, resulting in \(-1 \cdot x^{-2}\). In our example, this gives \(-x^{-2}\) when differentiating \( x^{-1} \). The power rule operates like a magic wand for derivatives, quickly turning the cumbersome process into an easily manageable one.

Exponent Properties

Exponent properties help in expressing functions in a form that is friendlier for differentiation. When dealing with exponents, you can switch expressions from division to multiplication by altering the exponent's sign. For example, \( \frac{1}{x} \) which is \( x^{-1} \), utilizes the property that \( x^{-n} = \frac{1}{x^n} \). This inverse transformation is essential because working with multiplication often simplifies differentiation over division. In the solution, \( \frac{e^{-2}}{x} \) was rewritten as \( e^{-2} \cdot x^{-1} \). This transformation not only utilized exponent properties but also paved the way for using differentiation rules effectively. Understanding and utilizing these easy yet powerful exponent properties is crucial for handling a wide variety of calculus problems smoothly.