Problem 27 - Calculus for Biology and Medicine

Question

Differentiate $$ h(s)=r s^{2}-r $$

Step-by-Step Solution

Answer

The derivative of \( h(s) = rs^2 - r \) is \( 2rs \).

Step 1: Identify the Function

The given function is \( h(s) = rs^2 - r \). We need to differentiate this with respect to \( s \).

Step 2: Apply the Power Rule

The power rule states that \( \frac{d}{ds}[s^n] = n s^{n-1} \). For the term \( rs^2 \), this becomes \( r \cdot 2s^{2-1} = 2rs \).

Step 3: Differentiate the Constant Term

The derivative of a constant \( r \) with respect to \( s \) is 0, since constant terms do not change with \( s \).

Step 4: Combine the Results

Combine the derivatives from Steps 2 and 3. The derivative of \( h(s) = rs^2 - r \) is \( 2rs + 0 = 2rs \).

Key Concepts

The Power Rule

In calculus differentiation, the power rule is a fundamental tool for finding derivatives quickly and efficiently. The power rule can be remembered with the formula:

• To differentiate a term of the form \( s^n \), bring the exponent \( n \) down in front of the term as a coefficient.
• Then, subtract one from the original exponent to find the new exponent of \( s \).

This translates to the mathematical expression \( \frac{d}{ds}[s^n] = n s^{n-1} \).
Let's apply this to a real example. Consider the term \( rs^2 \). By applying the power rule, first take the exponent 2 and bring it down to multiply the coefficient \( r \). This results in \( 2r \).
Then, subtract one from the exponent 2, resulting in \( s^{1} \) or simply \( s \). The differentiated term, therefore, becomes \( 2rs \). This is the essence of using the power rule for a straightforward and quick differentiation process.

Understanding Derivatives

Derivatives form the backbone of calculus by representing how a function changes. The derivative of a function provides the rate at which the function's value is changing at any given point. In simple terms, it's like figuring out the speed of a changing object.
When we differentiate a function like \( h(s) = rs^2 - r \), we are interested in understanding how \( h(s) \) behaves as \( s \) changes.

• The derivative gives us the slope of the tangent line at any point on the graph of the function.
• This slope shows whether the function is increasing or decreasing at that particular value of \( s \).

For example, in the function \( h(s) = rs^2 - r \), finding the derivative helps us see how the term \( rs^2 \) influences the overall behavior of the function as \( s \) varies. This insight is crucial for applications in physics, engineering, and economics, where understanding how quantities change is essential.

Constant Term Differentiation

In differentiation, we often encounter constant terms that remain unaffected by the variable in question. These constant terms are simple yet important in calculus. A constant is a number that does not change, no matter how the variable changes.
When differentiating, the derivative of a constant term with respect to any variable is always zero.

• This happens because the derivative measures change, and a constant does not change.
• Mathematically, it is expressed as \( \frac{d}{ds}[c] = 0 \), where \( c \) is a constant.

In our example function \( h(s) = rs^2 - r \), the term \( - r \) is a constant. Differentiating it results in zero. This is why, when we differentiated the entire function, the constant term did not contribute to the final derivative, which simplifies the differentiation process.
Understanding how to handle constants is essential because they often appear in functions that model real-world situations and knowing they result in zero simplifies calculations.