Problem 106 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=x^{7}+10$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 7x^6 \).

Step 1: Identify the function form

The function given is a polynomial function of the form \( f(x) = x^n + c \), where \( n = 7 \) and \( c = 10 \).

Step 2: Apply the power rule for differentiation

The power rule states that if \( f(x) = x^n \), then its derivative \( f'(x) = nx^{n-1} \). We apply this rule to \( x^7 \).

Step 3: Differentiate the constant

The derivative of a constant \( c \) is zero since constants do not change. Thus, the derivative of \( 10 \) is \( 0 \).

Step 4: Write the derivative

Combine the determined derivatives. Since the derivative of \( x^7 \) is \( 7x^6 \) and the derivative of \( 10 \) is \( 0 \), the derivative \( f'(x) \) becomes \( 7x^6 + 0 \).

Step 5: Simplify the expression

Eliminate any unnecessary terms (like zero) to obtain the simplest form of the derivative. So, \( f'(x) = 7x^6 \).

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression that consists of variables and coefficients. These are combined using only addition, subtraction, multiplication, and non-negative integer exponents of variables.
In simpler terms, a polynomial function is formed by summing terms of the form \(ax^n\), where \(a\) is the coefficient and \(n\) is a non-negative integer exponent.

• An example of a polynomial function is \(f(x) = x^7 + 10\).
• The highest exponent in the polynomial is called the degree. Here, the degree is 7.
• Polynomials can have multiple terms, but each part, like \(x^7\) or a constant \(10\), is also considered a term.

Understanding polynomial functions is essential because they are the backbone of many algebraic concepts and are key in differentiation problems.

Power Rule

The power rule is a simple yet powerful tool in calculus used to find the derivative of functions involving powers of \(x\). It allows us to efficiently differentiate terms like \(x^n\). The power rule states:

• If \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \).
• This means you multiply the original power \(n\) by the coefficient of \(x^n\) (which is often 1), and reduce the power by one.
• For instance, for \( x^7 \), applying the power rule gives \( 7x^6 \).

This rule greatly simplifies the process of differentiation, especially for polynomial terms. Remember, the power rule only applies to terms with variables raised to a power. Its simplicity lies in handling each term independently, which is helpful when dealing with longer polynomial expressions.

Constant

A constant in mathematics is a fixed value that does not change. When it comes to differentiation, constants have a unique property. Their derivative is always zero.
Why? Because differentiation measures how a function changes. Since a constant doesn't change, its rate of change is zero.In our example, \(10\) is a constant. Applying its rule:

• The derivative of any constant value \(c\) is \(0\).
• This is because constants do not vary as \(x\) changes, leading to no ‘slope’ or ‘rate of change’.

When you differentiate a polynomial function, identifying and correctly differentiating constants is crucial as it ensures accurate solutions. They contribute zero to the derivative expression.

Differentiation Steps

Differentiation is the process of finding the derivative of a function, indicating how it changes. With polynomials, the steps are systematic and predictable. Let's break it down as done in the given solution.

• **Identify Function Type**: Recognizing it as a polynomial helps plan the differentiation method. Here, \(f(x) = x^7 + 10\).
• **Apply the Power Rule**: For each term with \(x\), use the power rule. E.g., \(x^7\) becomes \(7x^6\).
• **Differentiate Constants**: As mentioned, the derivative of \(10\) is \(0\), because constants don’t change.
• **Combine Results**: Assemble differentiated terms. Here, the derivative becomes \(7x^6 + 0\).
• **Simplify**: Remove any terms that don’t affect the result, like zero, giving the final form \(f'(x) = 7x^6\).

These steps ensure clarity and efficiency by working through each component of the polynomial separately. This methodical process aids in handling increasingly complex polynomial functions in the future.