Problem 64 - Calculus for Biology and Medicine

Question

Differentiate with respect to the independent variable. $$ f(x)=x^{5}-\frac{1}{x^{5}} $$

Step-by-Step Solution

Answer

The derivative is \( 5x^4 + 5x^{-6} \).

Step 1: Identify the terms

The given function is \( f(x) = x^{5} - \frac{1}{x^{5}} \). This function consists of two separate terms: \( x^{5} \) and \( -\frac{1}{x^{5}} \).

Step 2: Differentiate the first term

The derivative of \( x^5 \) with respect to \( x \) is found using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). So, \( \frac{d}{dx}(x^5) = 5x^4 \).

Step 3: Differentiate the second term

Represent the second term \( -\frac{1}{x^5} \) as \( -x^{-5} \). Use the power rule to differentiate: if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Thus, \( \frac{d}{dx}(-x^{-5}) = (-1)(-5)x^{-5-1} = 5x^{-6} \).

Step 4: Combine the derivatives

Combine the derivatives of each term to get the derivative of the entire function. Therefore, \( \frac{df}{dx} = 5x^4 + 5x^{-6} \).

Step 5: Simplify if possible

The derivative \( 5x^4 + 5x^{-6} \) could be left as is, but it can also be rewritten to combine similar parts: \( = 5(x^4 + \frac{1}{x^6}) \).

Key Concepts

Power Rule

The power rule is a fundamental concept in differentiation, especially useful when working with polynomial functions. It simplifies finding derivatives by turning it into a systematic process. To apply the power rule, you must start by identifying the term of the function that looks like this: - If you have a term, say, \(x^n\), the power rule tells us that its derivative, with respect to \(x\), is \(nx^{n-1}\).This means we take the exponent \(n\), multiply it by the coefficient of \(x\), and then decrement the exponent by 1. For example:- Differentiating \(x^5\) using the power rule gives us \(5x^4\).An important detail to remember, especially when working with variables in the denominator, is to rewrite terms like \(\frac{1}{x^n}\) as negative exponents, such as \(x^{-n}\). This makes applying the power rule straightforward. As seen with \(-\frac{1}{x^5}\), we rewrite it as \(-x^{-5}\) and then differentiate to obtain \(5x^{-6}\).By mastering the power rule, you'll find that differentiation becomes a much quicker and easier task!

Differentiation

Differentiation is the process of finding the derivative of a function. This is crucial in calculus as it helps describe how a function changes at any point. When differentiating, it is important to respect the structure of the function and apply the rules accordingly. Here are some guiding principles:- Begin by identifying each term of the function. In our example, the original function was \(x^5 - \frac{1}{x^5}\).- Differentiate each term separately. For \(x^5\), using the power rule gives us \(5x^4\). Similarly, recognize \(-\frac{1}{x^5}\) as \(-x^{-5}\), then apply the power rule to get \(5x^{-6}\).- Combine the derivatives of each term to form the complete derivative. Thus, the derivative of the entire function is \(5x^4 + 5x^{-6}\).Differentiation allows us to analyze the rate of change and behavior of functions, which is foundational in understanding mathematical models and solving real-world problems.

Independent Variable

In calculus, an independent variable is one whose variation does not depend on that of another. Typically, it's the variable with respect to which we differentiate. In most functions, it's denoted by \(x\). Understanding its role is crucial:- The independent variable is what you manipulate or choose, and it determines the value of the function. In \(f(x) = x^{5} - \frac{1}{x^{5}}\), \(x\) is the independent variable.- When finding the derivative, you are looking at how changes in the independent variable \(x\) affect the function \(f(x)\).It is also important to note that choosing the correct independent variable is essential, as differentiation requires specifying it. For clarity, always verify which variable your function explicitly depends on, especially when working with multiple variables.