Problem 16 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\frac{1}{2} x^{2} e^{3}-x^{4} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = xe^3 - 4x^3 \).

Step 1: Apply the Power Rule and Constant Rule

To differentiate the function, we start by applying the Power Rule and Constant Rule. The function is given by:\[ f(x) = \frac{1}{2} x^{2} e^{3} - x^{4} \]The term \( e^3 \) is a constant, so we can treat \( \frac{1}{2} e^3 \) as a constant coefficient in the first term. Differentiating \( x^2 \) with respect to \( x \) gives \( 2x \). Hence, the derivative of the first term is:\[ \frac{d}{dx} \left( \frac{1}{2} x^2 e^3 \right) = \frac{1}{2} e^3 \cdot 2x = x e^3 \].

Step 2: Differentiate the Second Term Using the Power Rule

Now differentiate the second term \( -x^4 \) using the Power Rule. According to the Power Rule, if \( f(x) = x^n \) then \( f'(x) = nx^{n-1} \).Applying this rule:\[ \frac{d}{dx} (x^4) = 4x^3 \]Thus, the derivative of \( -x^4 \) is:\[ -4x^3 \].

Step 3: Combine the Results

Combine the derivatives of both terms to find the overall derivative of \( f(x) \):\[ f'(x) = x e^3 - 4x^3 \].

Step 4: Write the Final Derivative

After combining and simplifying, the final expression for the derivative of the function \( f(x) = \frac{1}{2} x^{2} e^{3} - x^{4} \) with respect to \( x \) is:\[ f'(x) = xe^3 - 4x^3 \].

Key Concepts

Power Rule

When you're dealing with algebraic expressions in calculus, the Power Rule is an essential tool for differentiation. It allows you to easily find the derivative of powers of variable terms, which are often present in functions. The rule states that if you have a term of the form \(x^n\), where \(n\) is any real number, the derivative is calculated by multiplying the power by the coefficient and then reducing the power by one.To break it down further, the Power Rule can be summarized in the formula:

• If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).

Here's why it works: By bringing down the exponent and multiplying, you're scaling the term according to the degree of power it initially had. This is why you see the power reducing by one unit in the new exponent. In our example, the power rule was applied to both terms: the \(x^2\) in the first term, and \(x^4\) in the second term.Applying this logic to the function given, you see how differentiating \(x^2\) turns it into \(2x\), and \(-x^4\) becomes \(-4x^3\). These transformations make taking derivatives straightforward and efficient.

Constant Rule

The Constant Rule is another fundamental concept in differentiation. Understanding this rule helps you work effectively with terms that have constants attached either by multiplication or themselves.Simply put, the Constant Rule states:

• The derivative of any constant is zero.
• If a constant is multiplied by a function, the constant can be "pulled out" of the derivative operation.

In our problem, the term \(e^3\) is a constant because the base "e" (the natural exponential base) raised to any power remains constant if the exponent does not change with respect to the variable. Here, it doesn't depend on \(x\). Thus, when you differentiate a term like \(\frac{1}{2}x^2e^3\), you treat \(\frac{1}{2}e^3\) as a constant. This allows the term \(x^2\) to be differentiated separately, simplifying the process significantly.This rule makes differentiation of terms with constants very manageable and helps break down the differentiation process into smaller, easier components.

Derivative

A derivative, in essence, is the measure of how a function changes as its input changes. It's like finding out the speed of a moving car at any given moment. The derivative provides the slope of the tangent line to the curve at any point, representing the rate of change.Here's how you can interpret derivatives in a simple way:

• Think of it as the 'instantaneous rate of change' or the velocity of change for a function.
• The derivative at a point tells you how steep or flat the curve is at that point.

In mathematical terms, if \(f(x)\) represents a function of \(x\), then \(f'(x)\) is the derivative of \(f(x)\) with respect to \(x\). In the provided exercise, the initial function \(f(x) = \frac{1}{2}x^2e^3 - x^4\) becomes \(f'(x) = xe^3 - 4x^3\), after applying the differentiation rules.Derivatives are pivotal in calculus, helping us understand change, optimize problems, or even solve physics equations involving motion or growth. Once you grasp the concept of derivatives and the rules that simplify finding them, tackling a wide array of problems in both math and applied sciences becomes much more manageable.