Problem 5 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=3-4 x-5 x^{2} $$

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = -4 - 10x \).

Step 1: Identify the Differentiation Rule

The function is a polynomial, and we need to differentiate it using the power rule. The power rule states that the derivative of \( ax^n \) is \( nax^{n-1} \).

Step 2: Differentiate Each Term

Apply the power rule to each term of the function: - The derivative of the constant term 3 is 0. - For the term \(-4x\), the derivative is \(-4\) as \(-4x^1\) becomes \(-4 \times 1 \times x^{1-1} = -4\).- For the term \(-5x^2\), the derivative is \(-10x\) as the power rule gives \(-5 \times 2 \times x^{2-1} = -10x\).

Step 3: Write the Derivative Function

Combine the derivatives of each term: Thus, the derivative of \( f(x) = 3 - 4x - 5x^2 \) is \( f'(x) = -4 - 10x \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They have the general form:

• \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants and \( n \) is a non-negative integer. Polynomials can be as simple as a constant, or they can include vast numbers of terms.

In the context of Calculus Differentiation, polynomial functions are significant because their derivatives can be calculated using simple rules like the Power Rule. Understanding the structure of polynomial functions - including terms, coefficients, and powers - is crucial for applying differentiation rules effectively and correctly.

Power Rule

The Power Rule is a fundamental tool in differential calculus used to find the derivative of polynomial functions. It states that if you have a function of the form \( ax^n \), its derivative is \( n \cdot ax^{n-1} \).

This rule simplifies the process of differentiation by providing a straightforward formula for handling polynomial terms. The process involves multiplying the coefficient \( a \) by the exponent \( n \) and then reducing the power of \( x \) by one.

• For a constant term, like 3 in our exercise, the derivative is 0. Constants have no variable to change with respect to, hence their rate of change is zero.
• For \( -4x \), considered as \( -4x^1 \), applying the Power Rule gives a derivative of -4.
• For \( -5x^2 \), the derivative, according to the Power Rule, is \(-10x\).

The Power Rule is efficient and easy, making it essential for simplifying derivatives of polynomial functions.

Derivative Calculation

Calculating derivatives is all about determining the rate of change of a function. In the context of polynomial functions, this requires applying rules of differentiation to each term of the function.

Let's review the steps of deriving the function \( f(x) = 3 - 4x - 5x^2 \):

• First, identify which rule applies: here, it's the Power Rule, because we're working with a polynomial.
• Differentiate each term individually using the Power Rule:
  • The constant 3 differentiates to 0.
  • The linear term \( -4x \) becomes \( -4 \).
  • The quadratic term \( -5x^2 \) turns into \( -10x \).
• Combine the derivatives: Hence, the derivative of the entire function is \( f'(x) = -4 - 10x \).

Understanding this step-by-step process is crucial for anyone working to master derivatives, as each step builds on the application of basic rules to more complex functions. Mastery of these techniques allows for estimating the behavior of functions reliably.