Problem 21 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\pi x^{3}-\frac{1}{\pi}+\frac{x}{\pi} $$

Step-by-Step Solution

Answer

The derivative of the function is \(f'(x) = 3\pi x^2 + \frac{1}{\pi}\).

Step 1: Express the Function

The given function is: \[ f(x) = \pi x^3 - \frac{1}{\pi} + \frac{x}{\pi} \] We need to differentiate this with respect to \(x\).

Step 2: Differentiate Each Term Separately

Differentiate each term in the function separately with respect to \(x\): 1. For the term \(\pi x^3\), use the power rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\). Differentiating \(\pi x^3\) gives: \[ \frac{d}{dx}(\pi x^3) = 3\pi x^2 \] 2. For the constant term \(-\frac{1}{\pi}\), the derivative is \(0\) because the derivative of a constant is zero.3. For the term \(\frac{x}{\pi}\), treat \(\frac{1}{\pi}\) as a constant coefficient. The derivative is: \[ \frac{d}{dx}\left(\frac{x}{\pi}\right) = \frac{1}{\pi} \]

Step 3: Combine the Derivatives

Add the results of the derivatives together to obtain the overall derivative of the function:\[ \frac{d}{dx}(f(x)) = 3\pi x^2 + 0 + \frac{1}{\pi} \]This simplifies to:\[ f'(x) = 3\pi x^2 + \frac{1}{\pi} \]

Key Concepts

Power Rule

The power rule is a fundamental technique in differentiation used to find the derivative of functions that involve variable terms raised to a power. This rule can be easily applied to monomials of the form \( x^n \) where \( n \) is any real number. The power rule states that if \( f(x) = x^n \), then its derivative is \( f'(x) = nx^{n-1} \). This simplifies the process of differentiation because once the exponent is known, you simply multiply the variable by this exponent and then subtract one from the exponent.

For example, when differentiating \( \pi x^3 \), you apply the power rule to get \( 3\pi x^2 \). Here:

• The exponent of the variable \( x \) is 3.
• According to the power rule: Multiply \( \pi \) (the coefficient) by 3 (the power).
• Subtract 1 from the power. The result is \( 3\pi x^2 \).

By using the power rule, you can effortlessly find the derivative of any standard polynomial term.

Derivative of a Constant

A constant is a term with a fixed value that doesn't change no matter the input. In differentiation, the derivative of a constant is always zero. This is because constants do not "move” along the x-axis; they have no rate of change.

When you are given a function, and you notice a constant term like \(-\frac{1}{\pi}\) in the example \( f(x) = \pi x^3 - \frac{1}{\pi} + \frac{x}{\pi} \), you identify \(-\frac{1}{\pi}\) as a pure constant. The derivative of this term is zero:

• Constants have no dependent variable (\(x\)) that they change with.
• Thus, \( \frac{d}{dx}(-\frac{1}{\pi}) = 0 \).

Recognizing constant terms and knowing their derivatives help simplify the differentiation process significantly.

Function of a Variable

A function of a variable indicates how the dependent variable changes as the independent variable varies. In differential calculus, you express this change by finding the derivative, which essentially provides the rate at which the function's value is changing.

In the function \( f(x) = \pi x^3 - \frac{1}{\pi} + \frac{x}{\pi} \), every term is interpreted based on how it incorporates the variable \( x \):

• \( \pi x^3 \) includes \( x \) with a power, making it a complete variable term which requires differentiation using the power rule.
• Each variable term with \( x \) shows the relation of the function to the variable itself.

By understanding how each component of a function relates to the variable, you can accurately apply differentiation rules to find how the function behaves as the variable changes. Differentiation exposes underlying patterns and allows us to predict and model behavior around certain inputs.