Problem 31 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative \( \frac{df}{dx} = 4ax \).

Step 1: Identify the Function

The function given to us is \( f(x) = 2a(x^2 - a) + a^2 \). Our goal is to differentiate this function with respect to \( x \). Notice that \( a \) is a constant.

Step 2: Simplify the Function

First, let's simplify the expression inside the brackets: \( f(x) = 2a(x^2) - 2a(a) + a^2 = 2ax^2 - 2a^2 + a^2 \). Combine like terms: \( f(x) = 2ax^2 - a^2 \).

Step 3: Differentiate Each Term

Differentiate the simplified function term by term:- Differentiate \( 2ax^2 \) with respect to \( x \). Use the power rule: \( \frac{d}{dx} [2ax^2] = 2a \cdot 2x = 4ax \).- Differentiate \( -a^2 \) with respect to \( x \). Since \( -a^2 \) is a constant, its derivative is \( 0 \).

Step 4: Combine the Derivatives

The derivative of the function \( f(x) = 2ax^2 - a^2 \) with respect to \( x \) is \( 4ax \) since the derivative of a constant is zero. Therefore, \( \frac{df}{dx} = 4ax \).

Key Concepts

Power Rule for Differentiation

The power rule is a fundamental concept in calculus, used for finding derivatives of functions of the form \[ x^n \]. This rule is an easy way to compute derivatives, making differential calculus less challenging.

If you have a function \[ f(x) = x^n \], the derivative is found by multiplying the exponent \( n \) by the coefficient in front of the term and then decreasing the exponent by one:

• Formula: \[ \frac{d}{dx}[x^n] = nx^{n-1} \]
• Example: The derivative of \( 3x^4 \) is \( 3 \times 4x^{3} = 12x^{3} \)

In our original exercise, we applied this rule to the term \( 2ax^2 \). Here, we considered \( a \) to be part of the coefficient. Therefore, the derivative is:\[ \frac{d}{dx}[2ax^2] = 2a \times 2x = 4ax \].

• Coefficient: \( 2a \)
• Exponent: \( 2 \)

By mastering the power rule, differentiating polynomial expressions becomes straightforward.

Constant Function Differentiation

Constant function differentiation is a simple concept but crucial in differentiating various expressions. When differentiating, constants behave in a specific predictable way – they vanish.

A constant is a value that does not depend on the variable being used for differentiation. When you differentiate a constant with respect to any variable, the result is always zero:

• Formula: \[ \frac{d}{dx}[c] = 0 \]
• Example: \( c = 5 \) with variable \( x \) gives \( \frac{d}{dx}[5] = 0 \)

In the exercise, \( -a^2 \) is considered constant because \( a \) is a constant independent of \( x \). Thus its derivative is:

\[ \frac{d}{dx}[-a^2] = 0 \].
This knowledge simplifies calculus, reducing our effort by focusing only on variable-dependent parts.

Simplifying Expressions for Differentiation

Before differentiating an expression, it’s often advisable to simplify it. This process ensures fewer errors and makes differentiation more manageable.

Simplification involves:

• Combining like terms.
• Distributing constants across terms in parentheses.
• Canceling out terms where possible.

In our task, we simplified:

• The original function: \( f(x) = 2a(x^2 - a) + a^2 \)
• To: \( 2ax^2 - 2a^2 + a^2 = 2ax^2 - a^2 \)

This reduction made the differentiation procedure straightforward, applying the power rule and constant differentiation effectively.
Simplifying expressions is an important step as it allows for clarity in mathematical processes, reducing the chances of mistakes.