Problem 47 - College Algebra and Calculus: An Applied Approach

Question

find \(f^{\prime}(x)\). $$ f(x)=x^{4 / 5}+x $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^{4 / 5}+x\) is \(f^{\prime}(x) = 4x^{-1 / 5} / 5 + 1\).

Step 1: Identify the function

The function given is \(f(x) = x^{4 / 5}+x\). This polynomial has two terms.

Step 2: Apply the power rule for first term

For the first term, which is \(x^{4 / 5}\), the power rule states that the derivative of \(x^n\) is \(n*x^{n-1}\). Applying this rule here, we get \((4 / 5)*x^{(4 / 5)-1}\), which simplifies to \(4x^{-1 / 5} / 5\).

Step 3: Apply the power rule for second term

For the second term, which is \(x\), applying the power rule gives us a derivative of 1, because the power of x in this case is 1, and subtracting 1 from it results in 0.

Step 4: Combine the results

Combine the derivatives of each term to get the derivative of the whole function. Therefore, \(f^{\prime}(x) = 4x^{-1 / 5} / 5 + 1\).

Key Concepts

Power Rule

In calculus, the power rule is a crucial method for finding the derivative of a term with a variable raised to a power. It's particularly helpful for functions consisting of polynomial terms. According to the power rule, if you have a function in the form of \(x^n\), its derivative is \(n \cdot x^{n-1}\). The process involves:

• Identifying the exponent \(n\) in the term \(x^n\).
• Multiply the exponent \(n\) by the base \(x\).
• Reduce the exponent by one, resulting in \(x^{n-1}\).

This rule streamlines differentiation, especially when dealing with multiple monomials. For example, when applied to the term \(x^{4/5}\), the power rule transforms it to \(\frac{4}{5} \cdot x^{-1/5}\). This simplification is what makes calculus more manageable.

Polynomials

Polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. A typical polynomial looks like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).In the context of differentiation, it's essential to recognize a function's polynomial structure. This recognition allows us to apply rules like the power rule and simplify the process of finding derivatives.

• Each term in a polynomial can be handled separately.
• Simplifies the calculation by following rules systematically.

For the given function \(f(x) = x^{4/5} + x\), the function consists of two terms: \(x^{4/5}\) and \(x\). Recognizing these components allows using specific calculus rules to find the derivative effectively, treating each term individually and ensuring accuracy in differentiation.

Differentiation

Differentiation is a process in calculus focusing on finding the rate at which a function changes at any given point, fundamentally represented by its derivative. The derivative of a function gives valuable insights into its behavior, such as the slope of its graph at specific points.The differentiation process for polynomial functions involves breaking down the components into manageable parts and applying calculus rules systematically. Steps to follow include:

• Identify each term in the function.
• Apply appropriate differentiation rules (e.g., power rule for polynomial terms).
• Combine the results to form the complete derivative expression.

For instance, in the exercise given, application of these steps leads to finding \(f'(x) = \frac{4}{5}x^{-1/5} + 1\). This result encapsulates the instantaneous rate of change for the original function \(f(x) = x^{4/5} + x\). Differentiation thus acts as a bridge to understanding the dynamic nature of mathematical functions.