Problem 21 - Calculus for Biology and Medicine

Question

Differentiate the functions given 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 \( f(x) = \pi x^3 - \frac{1}{\pi} + \frac{x}{\pi} \) is \( f'(x) = 3 \pi x^2 + \frac{1}{\pi} \).

Step 1: Identify the differentiation technique

The given function is a polynomial of the form \(f(x) = \pi x^3 - \frac{1}{\pi} + \frac{x}{\pi} \). We'll use the rule that the derivative of \(x^n\) is \(nx^{n-1}\), and the derivative of a constant is zero.

Step 2: Differentiate each term

Break down the function into its respective terms and find the derivative of each. - Derivative of the first term \( \pi x^3 \) is \( 3 \pi x^2 \) using the power rule. - Derivative of the second term \(-\frac{1}{\pi} \) is 0, as it is a constant.- Derivative of the third term \( \frac{x}{\pi} \) is \( \frac{1}{\pi} \) using the power rule on \( x^1 \).

Step 3: Combine the results

Add the derivatives of each term together to find the derivative of the entire function:\[f'(x) = 3\pi x^2 + 0 + \frac{1}{\pi}\]Simplifying this, the derivative of the function is:\[f'(x) = 3\pi x^2 + \frac{1}{\pi}\]

Key Concepts

Power Rule

The power rule is one of the fundamental rules you need to know in calculus for differentiation. It provides a quick and straightforward way to differentiate functions of the form \( x^n \), where \( n \) is any real number.

When applying the power rule, you follow a simple process:

• Multiply the exponent \( n \) by the coefficient of \( x^n \).
• Then, subtract one from the original exponent \( n \).

This yields the derivative: \( nx^{n-1} \). For example, if you have a term \( 5x^3 \), using the power rule, the derivative will be \( 15x^2 \).

In our original exercise, for the term \( \pi x^3 \), applying the power rule gives us \( 3\pi x^2 \). This is because we multiply \( 3 \) (the exponent) by \( \pi \) (the coefficient), and then subtract one from the exponent, resulting in \( 3\pi x^2 \). The power rule makes differentiating polynomial functions efficient and less overwhelming.

Polynomial Functions

Polynomial functions are expressions involving variables raised to non-negative integer powers with constant coefficients.

They take the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, ..., a_0 \) are constants and \( n \) is a non-negative integer.

• The function \( f(x) = \pi x^3 - \frac{1}{\pi} + \frac{x}{\pi} \) is a perfect example of a polynomial.
• It combines terms with different degrees: \( x^3 \), \( x^1 \), and a constant, \( -\frac{1}{\pi} \).

When differentiating polynomial functions, each term is differentiated separately, allowing you to apply differentiation rules like the power rule for each term independently.

Once each term is differentiated, the results are added together to form the derivative of the full polynomial. This method ensures that all aspects of the polynomial are considered, and it's a simple approach to solve complex polynomial derivatives.

Derivatives

Derivatives represent the rate at which a function is changing at any given point.

They are a basic tool in calculus and are essential for understanding how variables affect each other. Calculating derivatives involves finding an expression that describes this rate of change.

We denote the derivative of a function \( f(x) \) as \( f'(x) \). For a function like the one given in the original exercise, calculating its derivative involves using rules like the power rule. In this case:

• For \( \pi x^3 \), the derivative is \( 3\pi x^2 \), reflecting how changes in \( x \) affect this term.
• The constant \( -\frac{1}{\pi} \) disappears in differentiation because its rate of change is zero.
• The derivative of \( \frac{x}{\pi} \) is \( \frac{1}{\pi} \) since \( x^1 \) becomes \( 1x^0 = 1 \).

The combined derivative \( f'(x) = 3\pi x^2 + \frac{1}{\pi} \) gives a new function showing how \( f(x) \) changes with respect to \( x \). Derivatives are invaluable for analyzing and predicting the behavior of functions across different values, whether in physics, economics, or any other field that involves dynamic changes.