Problem 65 - Calculus Single Variable

Question

In Exercises 65-70, calculate the derivative of the given expression. $$ x^{-6} $$

Step-by-Step Solution

Answer

The derivative of \(x^{-6}\) is \(-6x^{-7}\).

Step 1: Identify the Function

The given expression is a single term, a power of x: \(x^{-6}\).

Step 2: Recall the Power Rule for Derivatives

The derivative of \(x^n\) with respect to \(x\) is \(n \cdot x^{n-1}\). This is known as the power rule for derivatives.

Step 3: Apply the Power Rule

For our function, \(n = -6\). Therefore, using the power rule, the derivative is \((-6) \cdot x^{-7}\).

Step 4: Simplify the Expression

The expression can be written as \(-6x^{-7}\), which is the final derivative.

Key Concepts

Power Rule

The Power Rule is a fundamental tool in calculus for finding derivatives of functions that are power of variables. Imagine you have a function in the form of \(x^n\), where \(n\) is any real number.
The Power Rule simplifies the derivative calculation by following a simple pattern:

• Multiply the exponent \(n\) by the coefficient of \(x\).
• Subtract one from the exponent \(n\).

So, if you start with a function \(f(x) = x^n\), the derivative \(f'(x)\) will be \(n \cdot x^{n-1}\). This simplifies complex calculus problems into straightforward calculations.
For example, given \(x^{-6}\), using the Power Rule, the derivative becomes \(-6x^{-7}\). This rule is most commonly applied in a step-by-step manner for effective differentiation.

Calculus

Calculus is a branch of mathematics focused on change. It’s composed of two main parts: differentiation and integration. Differentiation is all about finding how things change, while integration is about adding things up. Together, they create a powerful toolkit for mathematical modeling.
Differentiation specifically deals with the concept of the derivative, which tells us the rate of change of a function. This is crucial for many real-world applications, like understanding how quickly a car speeds up or slows down.
Studying calculus reveals the intricate dance of variables and equations. It’s a rite of passage for anyone delving into advanced mathematics, providing the language and tools to solve some of the most pressing scientific challenges.

Differentiation Techniques

Differentiation, or finding the derivative of a function, is a key skill in calculus. It requires using various techniques tailored to different types of functions.
Here are a few fundamental techniques:

• Power Rule: Useful for functions of the form \(x^n\).
• Product Rule: Used when differentiating products of two functions. If you have \(u(x) \cdot v(x)\), the derivative is \(u'(x) \cdot v(x) + u(x) \cdot v'(x)\).
• Quotient Rule: Applied when finding the derivative of a division of functions. For \(\frac{u(x)}{v(x)}\), the derivative is \(\frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{v(x)^2}\).
• Chain Rule: Essential for functions composed of other functions, like \(f(g(x))\). The derivative is \(f'(g(x)) \cdot g'(x)\).

Each technique helps break down complex derivatives into manageable steps. Mastering these moves empowers students to tackle diverse mathematical challenges with confidence.