Problem 23 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=a x^{3} $$ with respect to \(x\). Assume that \(a\) is a constant.

Step-by-Step Solution

Answer

The derivative is \( 3ax^2 \).

Step 1: Identify the function and the rule

The function to differentiate is given as \( f(x) = a x^3 \). We recognize this as a power function, which can be differentiated using the power rule. The power rule states that the derivative of \( x^n \) with respect to \( x \) is \( n x^{n-1} \).

Step 2: Apply the constant factor rule

Since \( a \) is a constant, we can apply the constant factor rule, which states that the derivative of a constant times a function is the constant times the derivative of the function. This means we keep \( a \) unchanged while differentiating \( x^3 \).

Step 3: Differentiate using the Power Rule

Using the power rule, differentiate \( x^3 \). The exponent is 3, so we bring down the exponent as a coefficient and decrease the exponent by one, giving us \( 3x^2 \). Thus, the derivative of \( x^3 \) is \( 3x^2 \).

Step 4: Combine the results

Multiply the constant \( a \) by the derivative we found in Step 3. This gives us \( a \cdot 3x^2 = 3ax^2 \). Therefore, the derivative of the function \( f(x) = ax^3 \) with respect to \( x \) is \( 3ax^2 \).

Key Concepts

Power Rule

The power rule is a fundamental rule in calculus used to differentiate functions of the form \( x^n \), where \( n \) is any real number. Let's break down how it works.

When applying the power rule, you start by identifying the exponent \( n \). The next step is to bring down this exponent as a coefficient in front of the variable \( x \). Finally, you subtract one from the original exponent to get the new exponent after differentiation.

With this simple three-step procedure:

• The original term is \( x^n \).
• After differentiation, it becomes \( nx^{n-1} \).

This method is straightforward and works for any polynomial function. For the expression \( x^3 \), by applying the power rule, the derivative comes out to\[3x^{2}\]Notice how the exponent 3 is moved to the front as a coefficient, and the power of \( x \) is reduced by one.

Constant Factor Rule

The constant factor rule is a handy tool in differentiation, especially when you encounter functions that have constants multiplied by a variable expression. It states that if you have a constant \( a \) multiplying a function \( u(x) \), the derivative is simply the constant times the derivative of that function.

Here's how to use it:

• Identify the constant. For a function like \( f(x) = ax^3 \), \( a \) is the constant factor.
• Take the derivative of the function without the constant. In this case, differentiate \( x^3 \) using the power rule, which gives \( 3x^2 \).
• Multiply the derivative by the constant. This results in \( a \cdot 3x^2 \), simplifying to \( 3ax^2 \).

The constant factor rule allows you to simplify the differentiation process by removing the constant from the operation initially and then reintroducing it after applying the derivative.

Derivative of Power Functions

Differentiating power functions is common in calculus, and it forms the basis for many other advanced concepts. A power function is any function that can be represented in the form \( ax^n \), where \( a \) is a constant and \( n \) is a real number.

When you differentiate a power function:

• Start by applying the power rule to the variable piece, \( x^n \).
• Utilize the constant factor rule if there's a constant, \( a \), involved in the expression, as in \( ax^n \).
• Combine the rules: The outcome for \( ax^n \) will be \( anx^{n-1} \), effectively merging the power and constant factor rules.

For the example \( f(x) = ax^3 \), using these steps, the derivative simplifies to \( 3ax^2 \). Each of these steps makes the process of finding derivatives fairly automatic for power functions, which is why learning these foundational rules is essential for becoming proficient in calculus.