Problem 9 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\left(x^{3}-\frac{7}{x}\right)^{-2} $$

Step-by-Step Solution

Answer

\( f^{\prime}(x) = -2 (x^3 - \frac{7}{x})^{-3} (3x^2 + \frac{7}{x^2}) \)

Step 1: Identify the Rule to Apply

The function is of the form \( (g(x))^n \) so we will apply the chain rule combined with the power rule, where \( f(x) = g(x)^n \) resulting in \( f^{\prime}(x) = n \cdot g(x)^{n-1} \cdot g^{\prime}(x) \).

Step 2: Differentiate Using the Power Rule

Apply the power rule to \( (x^3 - \frac{7}{x})^{-2} \). Using \( n = -2 \), the derivative is given by: \( -2 \cdot (x^3 - \frac{7}{x})^{-3} \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( g(x) = x^3 - \frac{7}{x} \). The derivative is \( g^{\prime}(x) = 3x^2 + \frac{7}{x^2} \).

Step 4: Combine the Results

Multiply the two derivatives from Step 2 and Step 3: \( f^{\prime}(x) = -2 \cdot (x^3 - \frac{7}{x})^{-3} \cdot (3x^2 + \frac{7}{x^2}) \).

Key Concepts

Chain Rule

Ever tried to simplify the process of unboxing nested gifts? When it comes to calculus, the chain rule is like finding the gift hidden inside layers of wrapping paper. The chain rule is a technique used to differentiate composite functions, those functions which have an outer and an inner function combined. In simpler terms, if you have a function that is the composition of two other functions, the chain rule helps us differentiate it in a structured manner. Let's break it down:

Imagine you have a function like \( f(x) = (g(x))^n \). Here, the outer function is something raised to the power of \(n\), and the inner function is \(g(x)\). With the chain rule, you first differentiate the outer function while keeping the inner function intact. Then, you multiply that result by the derivative of the inner function. This technique is essential for tackling complex derivatives efficiently. Next time you face a daunting composite function, remember the chain rule's elegance!

• Identify the outer and inner functions.
• Differentiate the outer function while keeping the inner function unchanged.
• Differentiating the inner function comes next.
• Multiply the derivatives obtained in the previous steps together.

Ensuring each step is clear will make your differentiation smooth.

Power Rule

Let's talk about the power rule, the bread and butter of differentiation techniques. It's as straightforward as slicing through butter with a warm knife! The power rule is a differentiation technique used for functions expressed as powers, like \( x^n \).

Here's the easy-peasy version: For any function that looks like \( x^n \), the derivative \( f'(x) \) is found by multiplying \( n \) by \( x^{n-1} \). Essentially, you bring down the power to the front as a multiplier and decrease the power by one. This rule is not just useful; it's fundamental for handling polynomials.

• For a simple equation \( f(x) = x^3 \), its derivative using power rule would be \( f'(x) = 3x^2 \).
• It works smoothly with negative and fractional exponents too.

Mastering the power rule simplifies the whole world of calculus, making it look less intimidating and more logical.

Differentiation Techniques

In calculus, differentiation techniques are like tools in a toolbox. Depending on the problem you’re dealing with, you’ll pick the appropriate tool. These techniques transform complex mathematical expressions into insights about change and rate.

We use a variety of techniques to differentiate functions based on their form:

• The **chain rule**, which we've discussed, for composite functions.
• The **power rule** for polynomials and similar expressions.
• The **product rule** and **quotient rule** for multiplying or dividing functions.
• Special techniques for trigonometric, exponential, and logarithmic functions.

Each technique tackles a specific form of function, making differentiation a customized process. Just like learning to use different tools effectively, mastering differentiation techniques provides a strong foundation for any calculus enthusiast. These methods work together, allowing us to approach any differentiation problem confidently and methodically.