Problem 54 - Calculus an Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{3}{\left(x^{3}-4\right)^{2}} $$

Step-by-Step Solution

Answer

The derivative of the given function is \(f'(x) = -18x^{2}(x^{3}-4)^{-3}\). The chain rule and power rule were used for this differentiation.

Step 1: Identify the Function Components

Recognize the function \(f(x) = \frac{3}{(x^{3}-4)^{2}}\) as a function of the form \(u^{-2}\), where \(u = x^3 - 4\).

Step 2: Applying Chain Rule and Power Rule

The chain rule states that if a variable \(z\) depends on the variable \(y\), which itself depends on the variable \(x\), that is, \(z = f(y)\) and \(y = g(x)\), then \(z\) can be expressed as a function of \(x\). Following this rule, you can write the derivative of \(f(x)\) as \(f'(x) = df/du * du/dx\).

Step 3: Compute \(df/du\)

Applying the power rule which states that, for any real number \(n\), the derivative of \(x^n\) is \(nx^{n-1}\), we can differentiate \(3u^{-2}\) to get \(df/du = -6u^{-3}\).

Step 4: Compute \(du/dx\)

Differentiate \(u = x^3 - 4\) with respect to \(x\) to get \(du/dx = 3x^{2}\).

Step 5: Substitute and Simplify

Substitute equations from steps 3 and 4 into equation from Step 2, i.e. \(f'(x) = df/du * du/dx = -6u^{-3} * 3x^{2} = -18x^2u^{-3}\). Now, replace \(u\) with original \(x^3 - 4\), the derivative is \(f'(x) = -18x^{2}(x^{3}-4)^{-3}\).

Key Concepts

Understanding the Chain Rule

When dealing with functions that are nested or composed, the chain rule is an essential tool in calculus differentiation. Imagine you have a function nested within another function, similar to our function in the problem: \(f(x)=3/(x^3-4)^2\). Here, the inner function is \(u = x^3 - 4\) and the outer function can be seen as \(u^{-2}\).
The chain rule helps us find how the change in \(x\) affects the change in the entire function \(f(x)\). This is achieved by breaking it down into two parts:

• First, find the derivative of the outer function with respect to the inner function (\(df/du\)).
• Then, find the derivative of the inner function with respect to \(x\) (\(du/dx\)).

The final step involves multiplying these two derivatives together to find \(f'(x): f'(x) = df/du \times du/dx\). This method simplifies differentiation, especially when functions appear complex and composed, facilitating more manageable calculations.

Exploring the Power Rule

In calculus differentiation, the power rule is a straightforward yet powerful tool. It concerns polynomials of the form \(x^n\), where \(n\) is a real number. The power rule states that the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\).
In the context of the example problem, the outer part of the function can be described using the power rule. Looking at \(u^{-2}\), where \(u = x^3 - 4\), applying the power rule gives \(-2u^{-3}\), which translates to multiplying by the exponent and reducing it by one.

• Step 1: Identify the exponent (which is \(-2\) in this instance).
• Step 2: Multiply the entire expression by this exponent \(-2\), resulting in \(-6u^{-3}\) after considering the constant \(3\) in the original function.
• Step 3: Reduce the exponent by one, changing \(-2\) to \(-3\) for the power of \(u\).

This method allows us to effectively manage polynomial expressions, even when intertwined with chains of functions, as seen with the interaction with the chain rule here.

Derivative of Polynomial Functions

Polynomial functions, like \(x^3 - 4\) in our problem, are frequently encountered in calculus and have a specific differentiation rule. Each term of a polynomial is its own standalone power function, making the power rule applicable for each term.
To differentiate such a polynomial:

• Differentiate each term individually. For \(x^3 - 4\), the derivative of \(x^3\) is \(3x^2\), using the power rule \((3x^{3-1})\).
• The derivative of the constant term \(-4\) is 0, as constants differentiate to zero because they don't change with \(x\).

Therefore, the derivative of the polynomial \(x^3 - 4\) is \(3x^2\). This process is systematic for handling the differentiations required in polynomial expressions, making it a fundamental concept for calculus students.