Problem 4 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=-3 x^{4}+6 x^{2}-2 $$

Step-by-Step Solution

Answer

The derivative is \(-12x^3 + 12x\).

Step 1: Identify the Function and Power Rule

We are given the function \( f(x) = -3x^4 + 6x^2 - 2 \). The task is to differentiate this function with respect to \( x \). We will use the power rule for differentiation, which states that if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).

Step 2: Differentiate Each Term

Apply the power rule term by term: 1. For \(-3x^4\), the derivative is \( -3 imes 4x^{4-1} = -12x^3 \).2. For \(6x^2\), the derivative is \( 6 imes 2x^{2-1} = 12x \).3. The derivative of a constant, \(-2\), is 0.

Step 3: Combine the Derivatives

Combine the derivatives of each term to find the derivative of the entire function:- The derivative of \(-3x^4\) is \(-12x^3\).- The derivative of \(6x^2\) is \(12x\).- The derivative of \(-2\) is 0.Thus, the derivative \( f'(x) \) is \(-12x^3 + 12x \).

Key Concepts

Understanding the Power Rule in Differentiation

The power rule is a key concept in calculus that makes finding the derivative of polynomial terms straightforward. When you have a function in the form of \( f(x) = ax^n \), the power rule helps you easily determine its derivative. By applying this rule, the derivative is given by \( f'(x) = anx^{n-1} \). Here's how it works:

• Identify the exponent (n): Look at the power of the variable \( x \) in each term of your function.
• Multiply by the exponent: Multiply the entire term by this exponent.
• Decrease the exponent by one: Reduce the exponent by one to find the new exponent for \( x \).

The power rule is particularly useful because it's quick and works directly on each term, making it ideal for differentiating polynomial functions.

What is a Polynomial Function?

Polynomial functions are algebraic expressions that involve sums of powers of a variable. These can be simple or complex, containing various terms with coefficients and exponents. A typical polynomial function looks like \( f(x) = ax^n + bx^{n-1} + \, \ldots \, + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( n \) is a non-negative integer.

• Degrees: The degree of a polynomial is the highest power of \( x \) present in the function. It tells you how many times polynomials can be multiplied to form the polynomial.
• Terms: Each part of the polynomial separated by a plus or minus sign is a term. For instance, \(-3x^4\), \(6x^2\), and \(-2\) are terms of the given polynomial.
• Coefficients: The numbers multiplying the powers of \( x \) are known as coefficients, such as -3, 6, and -2 in the example function.

Differentiating polynomial functions is often straightforward, thanks to the power rule, which can be applied term by term.

Steps in Derivative Calculation

Calculating the derivative of a polynomial function involves a systematic approach to differentiating each term. Let's break down how to compute the derivative of the example function \( f(x) = -3x^4 + 6x^2 - 2 \):

• Apply the power rule to each term: Take \(-3x^4\), for instance. Use the power rule: multiply -3 by 4 (the exponent of \( x \)), and reduce the exponent from 4 to 3. This gives \(-12x^3\).
• Repeat for other terms: For \(6x^2\), multiply 6 by 2, and then reduce the exponent of \( x \) by one, resulting in \(12x\).
• Derive constants directly: A constant like -2 has a derivative of 0 because constants do not change as \( x \) changes.
• Combine all derivatives: Put the results together to form the complete derivative of the polynomial function, which is \(-12x^3 + 12x\).

Using this method ensures accuracy and simplicity in obtaining the derivative of any polynomial.