Problem 47 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Functions to Differentiate

Our main function can be split into two sub-functions: \(x^{4 / 5}\) and \(x\). We will apply the power rule to both functions.

Step 2: Derive \(x^{4 / 5}\)

Using the power rule, the derivative of \(x^{4/5}\) is \(\frac{4}{5} x^{(4/5 - 1)}\). Reducing the exponent, we get: \(\frac{4}{5} x^{-1/5}\) or \( \frac{4}{5x^{1/5}}\).

Step 3: Derive \(x\)

The derivative of \(x\) is 1.

Step 4: Sum the Derivatives

The derivative of the sum of two functions is the sum of their derivatives. Therefore, the derivative of our original function \(f(x)=x^{4/5} + x\) is \(f^{\prime}(x) = \frac{4}{5x^{1/5}} + 1\).

Key Concepts

Power Rule

Understanding derivatives often starts with the Power Rule because it simplifies the process of differentiation significantly. The Power Rule is a fundamental technique in calculus, particularly useful when dealing with polynomial functions. This rule states that if you have a function of the form \(f(x) = x^n\), then its derivative \(f'(x)\) is \(n \cdot x^{n-1}\). Here's how it works:

• "\(n\)" is the exponent in the function.
• You multiply the function by this exponent.
• Then, you decrease the original exponent by one to find the new exponent for the differentiated term.

For instance, in the case of the function \(x^{4/5}\), applying the Power Rule means multiplying by \(4/5\) and reducing the power by 1, resulting in \(\frac{4}{5} x^{-1/5}\). This makes the Power Rule a quick and efficient way to find derivatives.

Differentiation

Differentiation is the process used in calculus to determine the rate at which a function is changing. It's the calculation of the derivative of a function. By finding a derivative, you're essentially finding a function that gives the slope of the tangent line at any point on the graph of the original function. This is extremely useful for

• Analyzing trends, such as increasing or decreasing behaviors in functions.
• Understanding how variables impact one another dynamically.

To differentiate a compound function, like \(f(x) = x^{4/5} + x\), you independently find the derivatives of its individual terms and then combine them. This process demonstrates the linearity property of differentiation: the derivative of a sum is just the sum of the derivatives. So, we find that \(f'(x) = \frac{4}{5}x^{-1/5} + 1\), neatly breaking down the changes for each part of the function.

Calculus

Calculus is a branch of mathematics focused on the study of change, utilizing concepts like derivatives and integrals. It's divided mainly into two areas: differential calculus, which concerns itself with rates of change and slopes of curves, and integral calculus, which involves the accumulation of quantities and the areas under curves.The significance of calculus is wide-reaching:

• It underpins the rules and equations used in physics, engineering, economics and more.
• It helps in predicting patterns and understanding the ever-changing world around us.

By mastering calculus and concepts like the Power Rule and differentiation, students can solve complex real-world problems and develop a deeper understanding of the mathematical descriptions of the universe. Every time you perform an operation such as finding \(f'(x)\) or apply rules like the Power Rule, you are engaging with calculus and its profound capabilities to model and analyze real-world dynamic systems.