Problem 7 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ g(s)=5 s^{7}+2 s^{3}-5 s $$

Step-by-Step Solution

Answer

The derivative is \( g'(s) = 35s^6 + 6s^2 - 5 \).

Step 1: Apply the Power Rule

The function to differentiate is \( g(s) = 5s^7 + 2s^3 - 5s \). For each term, apply the power rule for differentiation: the derivative of \( s^n \) is \( n \cdot s^{n-1} \).

Step 2: Differentiate Each Term

Differentiate each term separately:- The derivative of \( 5s^7 \) is \( 35s^6 \).- The derivative of \( 2s^3 \) is \( 6s^2 \).- The derivative of \( -5s \) is \( -5 \).

Step 3: Combine the Derivatives

Add up all the individual derivatives to get the final derivative of the function:\[ g'(s) = 35s^6 + 6s^2 - 5 \].

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus differentiation. It's often the very first technique taught to differentiate polynomial functions.
This rule states that for any term of the form \( s^n \), the derivative is \( n \cdot s^{n-1} \). Here’s how it works:

• Identify the exponent \( n \) of the variable \( s \).
• Multiply the entire term by the exponent \( n \).
• Reduce the exponent by one to get the new power of \( s \).

So, applying this to \( 5s^7 \), the power rule tells us to bring the 7 down in front to get \( 35 \) and then reduce the power to 6 to obtain \( 35s^6 \).
This consistent methodical approach makes the power rule easy to use for differentiating terms in a polynomial.

Derivative

A derivative represents the rate at which a function is changing at any given point. It's the foundation of calculus and an essential concept in understanding changes in mathematical models.
When finding the derivative of a polynomial function, you calculate the derivative of each term individually, using rules like the power rule.

• The derivative is often denoted with a prime symbol, as in \( g'(s) \).
• It helps evaluate the slope or steepness of the curve at every point.
• This slope can illustrate how rapidly a function is increasing or decreasing.

In the given problem, once each term is differentiated, you combine them to find the overall derivative \( g'(s) = 35s^6 + 6s^2 - 5 \).
This expression tells you exactly how the original function \( g(s) \) behaves when changing \( s \).

Polynomial Functions

Polynomial functions are expressions made up of variables raised to whole number powers, combined using addition, subtraction, and multiplication.
These functions form the building blocks of many mathematical equations and appear frequently in calculus problems.

• Terms in a polynomial generally appear in the format \( a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0 \).
• The highest degree of \( s \) determines the degree of the polynomial.
• Polynomials can describe various curves and shapes, depending on their coefficients and degrees.

The function \( g(s) = 5s^7 + 2s^3 - 5s \) is a 7th-degree polynomial due to the highest power, \( s^7 \).
Differentiating polynomials involves reducing the degree of each term by one, transforming curves into straighter lines at specific intervals.