Problem 1 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems 1-22 with respect to the independent variable. $$ f(x)=4 x^{3}-7 x+1 $$

Step-by-Step Solution

Answer

The derivative of the function is \(f'(x) = 12x^2 - 7\).

Step 1: Identify the Differentiation Rule

The function provided is a polynomial, specifically a cubic function, \[f(x) = 4x^3 - 7x + 1.\]To differentiate polynomials, use the power rule, which states that the derivative of \(ax^n\) is \(anx^{n-1}\).

Step 2: Differentiate Each Term

Apply the power rule to each term separately:- The derivative of \(4x^3\) is \(3 \times 4 x^{3-1} = 12x^2\).- The derivative of \(-7x\) is \(-7 \times 1 x^{1-1} = -7\).- The derivative of the constant \(1\) is \(0\).

Step 3: Write the Differentiated Function

Combine the differentiated results:\[f'(x) = 12x^2 - 7.\]

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in calculus that simplifies the differentiation of polynomial functions. When dealing with a term like \( ax^n \), the power rule states that its derivative is \( anx^{n-1} \). This rule is incredibly useful as it provides a quick way to find the derivative of terms with variables raised to a power.
For example, let's take the term \( 4x^3 \) from our original function. Here, \( a \) equals 4 and \( n \) equals 3.

• According to the power rule, bring down the exponent 3 in front of the term.
• Multiply the result, 3, by the coefficient 4 to get 12.
• Subtract one from the exponent, changing \( x^3 \) to \( x^2 \).

Thus, the derivative of \( 4x^3 \) becomes \( 12x^2 \). The power rule makes differentiation faster and is crucial for handling more complex expressions efficiently.

Cubic Functions Explained

Cubic functions, like \( f(x) = 4x^3 - 7x + 1 \), are polynomial expressions where the highest degree of the variable is three. These functions generally have an \( x^3 \) term as their leading term. Understanding the shape and behavior of cubic functions can be important in calculus and graphing.
The graph of a cubic function typically:

• Has an "S" shaped curve, featuring two bends or points of inflection.
• Can extend from negative to positive infinity or vice versa.
• Might intersect the x-axis at up to three points based on its roots.

For calculus, knowing that you're working with a cubic function helps when predicting the nature of its derivative. Differentiating reduces its degree by one, turning it into a quadratic, which can tell us more about the slope and concavity of the function at various points.

Polynomial Differentiation Steps

Differentiating a polynomial involves breaking the function down into individual terms and applying the differentiation rules, step-by-step. Polynomials consist of multiple terms, each involving variables raised to a power. This process is simplified by applying the power rule individually to each term.
Follow these steps to differentiate a polynomial:

• Identify each term in the polynomial and the power of its variable.
• Apply the power rule to each term. If a term is constant, its derivative is simply zero.
• Combine all the differentiated terms to form the derivative of the complete function.

For the function \( f(x) = 4x^3 - 7x + 1 \), we:

• Applied the power rule to \( 4x^3 \) to get \( 12x^2 \).
• Did the same for \( -7x \), resulting in \( -7 \).
• Took the derivative of the constant 1 as zero.
• Combined them to find that \( f'(x) = 12x^2 - 7 \).

Polynomial differentiation is foundational in calculus and offers clear insight into the behavior and characteristics of functions.