Problem 3 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable.$$ f(x)=-2 x^{5}+7 x-4 $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -10x^4 + 7 \).

Step 1: Identify the Problem

We need to find the derivative of the function \( f(x) = -2x^5 + 7x - 4 \) with respect to \( x \).

Step 2: Apply the Power Rule to Each Term

The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). We apply this rule to each term of the function separately.

Step 3: Differentiate the First Term

For the term \( -2x^5 \), the derivative is \( -2 \times 5x^{5-1} = -10x^4 \).

Step 4: Differentiate the Second Term

For the term \( 7x \), the derivative is \( 7 \times 1x^{1-1} = 7 \).

Step 5: Differentiate the Constant Term

For constant term \( -4 \), the derivative is \( 0 \) because the derivative of any constant is zero.

Step 6: Combine the Results

Combine the derivatives from each term: \( f'(x) = -10x^4 + 7 \).

Key Concepts

Power Rule

The power rule is a cornerstone concept in calculus, especially when dealing with polynomial functions. This rule simplifies the process of differentiation significantly. It provides a straightforward method for taking the derivative of any term of the form \( x^n \).
Consider the term \( x^n \). According to the power rule, when differentiating, you bring down the exponent \( n \) as a coefficient, and then subtract one from the exponent. The formula can be written as:

• \( \frac{d}{dx}[x^n] = nx^{n-1} \)

For example, in the function \( -2x^5 \), applying the power rule gives \(-2 \times 5x^{5-1} = -10x^4\). Each term in a polynomial can be handled individually in this manner, simplifying a potentially complex operation into a series of predictable steps.

Differentiate Functions

Differentiating functions involves finding the derivative of a mathematical expression. The derivative measures how a function changes as its input changes, providing a mathematical way to capture the rate of change or slope of the function at any point.
For polynomial functions, this involves applying the power rule to each term. In the given function \( f(x) = -2x^5 + 7x - 4\), the derivative is calculated by differentiating each term separately:

• For \( -2x^5 \), the derivative is \(-10x^4\).
• For \( 7x \), since \( x \) is the first power, you apply the power rule: \( 7 \times 1x^{1-1} = 7\).
• For constants like \(-4\), the derivative is zero because constants do not change.

By dealing with each term individually, you ensure that every aspect of the function's behavior is captured in its derivative.

Independent Variable Differentiation

When discussing differentiation, it's crucial to define with respect to what variable you are differentiating. This is often called the 'independent variable'. In most single-variable calculus, this is typically represented by \( x \). Differentiating with respect to the independent variable involves looking at how changes in this variable influence the entire function.
For example, in \( f(x) = -2x^5 + 7x - 4\), \( x \) is the independent variable. Each term of the function is differentiated concerning \( x \) according to the power rule. This approach ensures that any change in \( x \) results in a predicted change in \( f(x) \), as captured by \( f'(x) \).
Understanding which variable is independent helps in applying the correct differentiation rules and interpreting the resulting derivative correctly.