Problem 20 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\frac{x^{3}}{15}-\frac{x^{4}}{20}+\frac{2}{15} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{5}x^{2} - \frac{1}{5}x^{3} \).

Step 1: Differentiate the first term

The first term is \( \frac{x^{3}}{15} \). Using the power rule for differentiation, which states that \( \frac{d}{dx} [x^{n}] = n \cdot x^{n-1} \), we find the derivative to be \( \frac{3}{15} \cdot x^{2} = \frac{1}{5}x^{2} \).

Step 2: Differentiate the second term

The second term is \( -\frac{x^{4}}{20} \). Applying the power rule again, we get the derivative as \( -\frac{4}{20} \cdot x^{3} = -\frac{1}{5}x^{3} \).

Step 3: Differentiate the constant term

The last term is \( \frac{2}{15} \), which is a constant. The derivative of a constant is 0.

Step 4: Combine the derivatives

Add the derivatives from Steps 1, 2, and 3: \( \frac{1}{5}x^{2} - \frac{1}{5}x^{3} + 0 \). This simplifies to \( \frac{1}{5}x^{2} - \frac{1}{5}x^{3} \).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus, especially when dealing with polynomials. Its formula can be summed up simply: if you have a function in the form of \( x^n \), then its derivative is \( n \cdot x^{n-1} \). This rule allows for quick differentiation, saving you from complex calculations for each term that follows this pattern.

Let's break it down further:

• Identify the power of the variable \( x \) in each term.
• Multiply this power by the coefficient (the number in front of \( x^n \)).
• Reduce the power of \( x \) by one.

The power rule is one of the first rules you will master in calculus because of its simplicity and versatility in handling polynomial expressions, similar to the ones seen in our step-by-step solution.

Calculus for Biology

While calculus might seem purely mathematical, it is highly relevant in various fields like biology. "Calculus for biology" involves applying calculus concepts, such as differentiation, to biological models and studies.

For example, biologists may use derivatives to model:

• Rates of population change, where the derivative describes changes in population over time.
• Enzyme activity rates, where differentiation is used to find rates of reaction based on substrate concentration.

Understanding calculus helps biologists predict changes, analyze biological processes, and model complex systems effectively. It's a bridge between mathematical abstraction and practical application in understanding the living world.

Derivative of Polynomial Functions

Derivatives of polynomial functions are straightforward due to the power rule. Polynomial functions consist of terms with variables raised to whole-number powers, often combined with constants. The straightforward application of the power rule makes these derivatives easy to calculate.

In working with polynomials:

• Differentiating each term individually is key.
• The derivative of a constant term in the polynomial is always zero since constants don't change.
• Summing the derived terms gives the overall derivative of the polynomial function.

This process simplifies many calculus problems, reduces the complexity of equations, and is often a first step in tackling more intricate calculus topics like integrals and differential equations.