Problem 34 - Calculus for Biology and Medicine

Question

Differentiate $$ h(s)=a^{4} s^{2}-a s^{4}+\frac{s^{2}}{a^{4}} $$ with respect to \(s\). Assume that \(a\) is a positive constant.

Step-by-Step Solution

Answer

The derivative is: \(h'(s) = 2a^4 s - 4a s^3 + \frac{2s}{a^4}\).

Step 1: Differentiate the First Term

The first term of the function is \(a^4 s^2\). To differentiate this with respect to \(s\), multiply the coefficient \(a^4\) by the power of \(s\) (which is 2), and then reduce the power of \(s\) by one. This gives: \(2a^4 s\).

Step 2: Differentiate the Second Term

The second term is \(-a s^4\). Differentiate this by multiplying \(-a\) by the power of \(s\) (which is 4), and then reduce the power of \(s\) by one. This results in: \(-4a s^3\).

Step 3: Differentiate the Third Term

The third term is \(\frac{s^2}{a^4}\). Treat this as \(\frac{1}{a^4} s^2\). To differentiate, multiply \(\frac{1}{a^4}\) by the power of \(s\) (which is 2), and then reduce the power of \(s\) by one. This yields: \(2\frac{s}{a^4}\).

Step 4: Combine the Derivatives

Now, combine the results from the differentiation of each term. The derivative of \(h(s)\) is: \(h'(s) = 2a^4 s - 4a s^3 + \frac{2s}{a^4}\).

Key Concepts

Understanding Calculus in Differentiation

Calculus is a branch of mathematics focused on rates of change and accumulation. One of the most significant aspects of calculus is differentiation. Differentiation allows us to determine how a function changes at any given point and is instrumental in finding the slope of a curve.
To grasp differentiation effectively, it's important to understand its relationship to calculus as a whole:

• In essence, differentiation refers to finding the derivative of a function. This is a measure of how a function’s output value changes as its input changes.
• Differentiation is used to solve problems involving rates of change, such as velocity in physics or growth rates in biology.
• It involves various rules and techniques to find derivatives, including the power rule, product rule, and chain rule, each with their specific applications.

With differentiation, we explore not just the value of functions but understand how those values change—a fundamental part of how we perceive and apply mathematical concepts in the real world.

Exploring Polynomial Differentiation

Polynomials are algebraic expressions made up of terms that are added or subtracted. Each term consists of a coefficient and a variable raised to a power. Polynomial differentiation refers to the process of finding the derivative of such expressions.
Working with polynomials is usually straightforward thanks to their structure. Here's what to keep in mind:

• Since polynomials are simple sums of power terms, their differentiation involves applying rules like the power rule to each term individually.
• Polynomials could have varying powers and coefficients, which can be constants or variables.
• Understanding the behavior of these expressions, which often involves using polynomial differentiation, is crucial in graphical analysis and solving real-life problems.

When differentiating polynomials, you'll notice repeated applications of basic differentiation rules, making it a perfect way to get comfortable with calculus concepts.

Mastering the Power Rule in Differentiation

The power rule is one of the most frequently used and simplest techniques in differentiation, especially when dealing with polynomial expressions. It provides a straightforward method for finding the derivatives of functions where the variables are raised to a power.
Applying the power rule involves a simple process:

• For any term of the form \(x^n\), multiply by the power \(n\) and decrease the power by one, yielding \(nx^{n-1}\).
• This rule is applicable to expressions with constants as well, such as \(kx^n\), where \(k\) is a constant. The derivative becomes \(knx^{n-1}\).
• The power rule can also be applied to negative and fractional powers, making it a versatile tool.

By mastering the power rule, you can handle a variety of differentiation tasks with ease, particularly those involving polynomial terms, thus making it an essential part of calculus learning.