Problem 29 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=r s^{2} x^{3}-r x+s $$

Step-by-Step Solution

Answer

The derivative of the function is \(3rs^2 x^2 - r\).

Step 1: Recognize the Differentiation Rule

To differentiate the function, recall that the derivative of a sum is the sum of the derivatives. We will apply the power rule and constant rule of differentiation to the function \(f(x) = rs^2 x^3 - rx + s\).

Step 2: Differentiate Each Term Individually

First, differentiate \(rs^2 x^3\). Using the power rule, where \(\frac{d}{dx} x^n = nx^{n-1}\), the derivative is \(3rs^2 x^2\). Next, consider \(-rx\), which differentiates to \(-r\) since \(\frac{d}{dx} x = 1\). The constant \(s\) differentiates to \(0\) since constants have a derivative of zero.

Step 3: Sum Up the Derivatives

Now, add up all the derivatives from Step 2: The derivative of \(rs^2 x^3\) is \(3rs^2 x^2\), the derivative of \(-rx\) is \(-r\), and the derivative of \(s\) is \(0\). Thus, the derivative of the entire function is \(3rs^2 x^2 - r\).

Key Concepts

Power Rule

The power rule is one of the fundamental techniques in calculus used for differentiation. It provides a straightforward way to find the derivative of a power function, which is any function of the form \( f(x) = x^n \). Here, \( n \) is a real number. The power rule states:

• If \( f(x) = x^n \), then the derivative \( f'(x) = n x^{n-1} \).

This rule tells us that we can differentiate \( x^n \) by multiplying the exponent \( n \) with \( x^{n-1} \), which effectively reduces the exponent by one. In the exercise, the term \( rs^2 x^3 \) was differentiated using the power rule:

• Here, \( n = 3 \), so the derivative is \( 3(rs^2) x^{3-1} \), simplifying to \( 3rs^2x^2 \).

By applying this rule, we efficiently compute the derivative of any term with a variable raised to a power.

Constant Rule

The constant rule is another simple but essential rule in differentiation. It deals with terms in a function that are constant – meaning they do not change as \( x \) changes. The constant rule can be stated as follows:

• The derivative of a constant is zero.

This means that any term in a function that does not contain the variable \( x \) will have a derivative of zero. In the given exercise, the term \( s \) is constant:

• When we differentiate \( s \), the result is zero.

Applying this rule simplifies calculations and helps remove constant terms when computing the derivative of a function.

Derivatives

Understanding derivatives is key to mastering calculus. A derivative represents the rate at which a function is changing at any given point, which can be thought of as the slope of the tangent line to the function's graph at that point.To differentiate a function means to find its derivative. Here are some important points about derivatives:

• They provide information about the behavior of functions, such as whether it is increasing or decreasing.
• The process of differentiation can be performed on each term of a function separately when applying rules such as the power and constant rules.

In our exercise, we differentiated each term of \( f(x) = rs^2 x^3 - rx + s \) separately:

• \( rs^2 x^3 \) differentiated to \( 3rs^2 x^2 \)
• \(-rx\) differentiated to \(-r \)
• The constant \( s \) differentiated to \( 0 \)

Summing these, we find that the derivative of the entire function is \( 3rs^2 x^2 - r \). Understanding this step-by-step process helps in grasping the concept of how derivatives capture the change in functions across different scenarios.