Problem 3 - The Calculus with Analytic Geometry

Question

Differentiate the given function by applying the theorems of this section. $$ f(x)=\frac{1}{8} x^{8}-x^{4} $$

Step-by-Step Solution

Answer

The derivative of \(f(x) = \frac{1}{8} x^8 - x^4\) is \(f'(x) = x^7 - 4x^3\).

Step 1: Identify the Power Rule

The power rule states that the derivative of \(ax^n\) is \(anx^{n-1}\). We will use this rule to differentiate each term of \(f(x)\) separately.

Step 2: Apply the Power Rule to the First Term

The first term is \(f_1(x) = \frac{1}{8}x^8\). Using the power rule, the derivative is: \(f_1'(x) = \frac{1}{8} \times 8 x^{8-1} = x^7\)

Step 3: Apply the Power Rule to the Second Term

The second term is \(f_2(x) = -x^4\). Using the power rule, the derivative is: \(f_2'(x) = -4 x^{4-1} = -4x^3\)

Step 4: Combine the Results

Now, add the derivatives of each term together to get the final derivative of the function: \(f'(x) = f_1'(x) + f_2'(x) = x^7 - 4x^3\)

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus. It allows us to find the derivative of any polynomial term with ease. The power rule states that if you have a term in the form of \(ax^n\), its derivative is given by: \[\frac{d}{dx}[ax^n] = anx^{n-1}\]
This rule is powerful because it simplifies the process of differentiation for terms with exponents. For example, if you start with \(x^3\), using the power rule gives you: \(3x^2\). The coefficient and the exponent both play crucial roles in applying this rule.

Derivative

A derivative represents the rate at which a function is changing at any given point. In simpler terms, it measures how a function's output value changes in response to small changes in its input value.
Derivatives are everywhere in calculus and have applications in various fields such as physics, engineering, and economics. They help in understanding things like acceleration, profit maximization, and even in predicting future trends.
For our function \(f(x)=\frac{1}{8} x^{8}-x^{4}\), we need to find the derivative to understand how this function behaves.

Function Differentiation

Function differentiation involves finding the derivative of a function. This process uses the rules of differentiation, such as the power rule. When differentiating a function with multiple terms, like \(f(x)\), it's essential to differentiate each term separately. Then, combine the results to find the overall derivative.
Let's differentiate our function \(f(x)\):

• First term: \( \frac{1}{8} x^8 \), using the power rule, we get \( x^7 \).
• Second term: \(-x^4\), using the power rule, we get \(-4x^3\).

Combining them, we have the final derivative: \(f'(x) = x^7 - 4x^3\).

Algebraic Operations

Algebraic operations are the basic mathematical operations like addition, subtraction, multiplication, and division. In calculus, these operations are often used after applying differentiation rules.
For the given function \(f(x)=\frac{1}{8} x^{8}-x^{4}\), after using the power rule for derivatives on each term, we use algebraic operations to combine the results.
So, after differentiating \(\frac{1}{8} x^8\) to get \(x^7\) and \(-x^4\) to get \(-4x^3\), we add them together: \(f'(x) = x^7 - 4x^3\).
This simplification is a fundamental part of solving calculus problems efficiently.