Problem 5 - Calculus for Biology and Medicine

Question

Differentiate the functions given 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 Function to Differentiate

The given function is \[ f(x) = 3 - 4x - 5x^2 \]We will differentiate this with respect to \(x\).

Step 2: Differentiate Constant Term

The derivative of a constant is 0. Hence, the derivative of the constant term \(3\) is \(0\).

Step 3: Differentiate the Linear Term

Use the power rule for differentiation, which states that the derivative of \(ax^n\) is \(anx^{n-1}\).So, the derivative of \(-4x\) is \[-4 \times 1 \times x^{1-1} = -4\]

Step 4: Differentiate the Quadratic Term

Again, using the power rule, differentiate \(-5x^2\):\[-5 \times 2 \times x^{2-1} = -10x^{1} = -10x\]

Step 5: Combine All Derivatives

Add all the derived components together:The derivative of the constant term \(0\), the linear term \(-4\), and the quadratic term \(-10x\).The result is: \[ f'(x) = 0 - 4 - 10x \]

Step 6: Simplify the Expression

Add the components together to write the final derivative:\[ f'(x) = -4 - 10x \]

Key Concepts

Understanding the Power Rule

When you're learning differentiation, the power rule will quickly become one of your best friends. The power rule is a shortcut used to differentiate functions of the form \(ax^n\), where \(a\) is a coefficient, and \(n\) is the exponent. The formula states that the derivative of \(ax^n\) is \(anx^{n-1}\).

Here's how it works: You multiply the entire term by the exponent, reduce the exponent by one, and adjust your term accordingly.

• Example: Differentiating \(5x^3\) would give \(5 imes 3 imes x^{3-1} = 15x^2\).

Throughout the differentiation process of any polynomial, employing the power rule is essential to simplifying and solving the problem efficiently.

Differentiating a Constant Term

Differentiating a constant may seem very straightforward, but it's crucial to understand how this works in calculus. The derivative of a constant is always zero. Why? Because a constant signifies a flat, unchanging slope when plotted as a function.

So, when you differentiate a constant term, like the number \(3\), its rate of change is nonexistent.

• In our exercise involving \(f(x) = 3 - 4x - 5x^2\), the derivative of the constant term \(3\) is simply \(0\).

Dropping these constant derivatives helps in focusing on changes depicted by variable terms.

Mastering Linear Term Differentiation

Differentiating a linear term is also simplified by applying the power rule. Linear terms have the general form \(ax\), effectively \(ax^1\). By using the power rule, we differentiate it as if it were any term with an exponent. The derivative of \(ax\) is simply \(a\).

In our example:

• The linear term \(-4x\) was differentiated to become \(-4\). This calculation stems from treating \(-4x\) as \(-4x^1\), then: \(-4 \times 1 \times x^{1-1} = -4\).

This rule elegantly highlights how each increase in input results in a constant output change, characteristic of linear functions.

Quadratic Term Differentiation

When dealing with quadratic terms, the power rule still holds supreme. Quadratic terms are expressed in the form \(ax^2\), and their differentiation follows the same pattern as other polynomial terms. The derivative here becomes more pronounced due to the higher exponent.

Following our example:

• By differentiating \(-5x^2\), we use the power rule to get \(-5 \times 2 \times x^{2-1} = -10x\).

Quadratic differentiation highlights the increasing rate of change, as shown in the derivative, compared to linear expressions. It provides invaluable insights, especially when assessing motion or growth behavior in various contexts.