Problem 26 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=a^{2} x^{4}-2 a x^{2} $$ with respect to \(x\). Assume that \(a\) is a constant.

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 4a^2 x^3 - 4a x \).

Step 1: Identify the Function to Differentiate

Examine the function that needs differentiation. The given function is \( f(x) = a^2 x^4 - 2a x^2 \), where \( a \) is a constant.

Step 2: Apply Power Rule to Each Term

Differentiate each term of the function separately. The power rule states that the derivative of \( x^n \) is \( n x^{n-1} \).

Step 3: Differentiate the First Term

For the term \( a^2 x^4 \), apply the power rule: the derivative is \( 4a^2 x^3 \).

Step 4: Differentiate the Second Term

For the term \( -2a x^2 \), apply the power rule: the derivative is \(-4a x \).

Step 5: Combine the Differentiated Terms

Combine the results from Step 3 and Step 4 to get the derivative of the entire function. Thus, the derivative \( f'(x) \) is \( 4a^2 x^3 - 4a x \).

Key Concepts

Power Rule

The power rule is a straightforward and powerful technique in calculus used for differentiation. It allows us to find the derivative of functions that involve power terms. The rule states that if you have a function in the form of \( x^n \), the derivative is calculated as \( n x^{n-1} \). This means you essentially bring down the power as a coefficient in front and then subtract one from the original power.

For example, in the function \( f(x) = x^4 \), the power rule tells us that its derivative \( f'(x) \) is \( 4x^3 \). It simplifies the process of differentiation by transforming power terms without needing complicated algebra.

• Useful for polynomial terms
• Simplifies the calculation of derivatives
• Works seamlessly with constants when multiplied with power terms

The power rule is essential for finding the derivatives of many algebraic functions and is the first rule you'll learn when starting with differentiation.

Differentiation

Differentiation is the process of finding the derivative of a function. It's a key concept in calculus that measures how a function's output value changes as its input value changes. In simple terms, it helps to find the rate of change or the slope of a function at any given point.

When differentiating a polynomial function like \( f(x) = a^2 x^4 - 2a x^2 \), each term is treated separately. The power rule, among other rules, helps facilitate this process. The derivative indicates how the original function, \( f(x) \), changes concerning \( x \).

Here’s the step-by-step process that involves differentiation:

• First, identify each term in the function that involves a variable \( x \).
• Apply differentiation rules such as the power rule to each term separately.
• Combine all differentiated terms to form the derivative of the entire function.

The power rule is particularly helpful in simplifying differentiation, especially for polynomial equations.

Calculus

Calculus is a branch of mathematics that focuses on studying change. It includes two major branches: differentiation and integration. Differentiation deals with finding the derivative, which represents the rate of change or the slope of a curve at any given point. On the other hand, integration is related to finding areas under curves and accumulated quantities.

In the context of this exercise where you differentiate \( f(x) = a^2 x^4 - 2a x^2 \), the use of calculus tools, particularly differentiation, is clearly demonstrated. Calculus provides the mathematical language to describe changes, whether related to movement, growth, or other dynamic processes, using functions.

• Differentiation and integration are the two core areas of calculus.
• It applies to various fields like physics, engineering, and economics.
• Calculus helps solve complex problems related to change and motion.

Calculus is fundamental in both theoretical math and practical applications, offering solutions to real-world problems through an understanding of change and motion.