Problem 342 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=x^{2} \ln 9 x$$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 2x \ln(9x) + x\).

Step 1: Identify the Product Rule

In the function \(f(x) = x^2 \ln(9x)\), we can see that it is a product of two functions: \(u(x) = x^2\) and \(v(x) = \ln(9x)\). We will apply the product rule for differentiation, which states that if \(f(x) = u(x)v(x)\), then \(f'(x) = u'(x)v(x) + u(x)v'(x)\).

Step 2: Differentiate \(u(x)\)

Find the derivative of \(u(x) = x^2\). The derivative of \(x^2\) with respect to \(x\) is \(u'(x) = 2x\).

Step 3: Differentiate \(v(x)\)

Find the derivative of \(v(x) = \ln(9x)\). Using the chain rule, we differentiate \(\ln(9x)\) as \(v'(x) = \frac{1}{9x} \cdot 9 = \frac{1}{x}\).

Step 4: Substitute into the Product Rule Formula

Substitute \(u(x)\), \(u'(x)\), \(v(x)\), and \(v'(x)\) into the product rule \(f'(x) = u'(x)v(x) + u(x)v'(x)\): \[ f'(x) = (2x)\ln(9x) + (x^2) \left(\frac{1}{x}\right) \]

Step 5: Simplify the Expression

Simplify the expression obtained from the product rule: \[ f'(x) = 2x \ln(9x) + x \].

Key Concepts

Product Rule

The product rule is an essential technique in calculus used when finding the derivative of a product of two functions. If you have a function characterized by the multiplication of two distinct functions, like \(f(x) = u(x)v(x)\), the product rule states:

• \(f'(x) = u'(x)v(x) + u(x)v'(x)\)

Essentially, it involves taking the derivative of the first function \(u(x)\), multiplying it by the second function \(v(x)\), then adding the product of the first function \(u(x)\) with the derivative of the second function \(v(x)\).
This method helps in effectively breaking down more cumbersome derivatives into manageable parts, making it much easier to solve. In our example function \(f(x) = x^2 \ln(9x)\), we used the product rule because it comprises a product of \(u(x) = x^2\) and \(v(x) = \ln(9x)\). By following the product rule, we differentiated each part accordingly.

Chain Rule

The chain rule is applied when dealing with composite functions, functions made up of one function inside another. It's useful for finding derivatives when functions involve nested terms. If you have a function of the form \(y = g(f(x))\), the chain rule tells us how to differentiate it:

• \(\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\)

In simple terms, differentiate the outer function \(g\), then multiply it by the derivative of the inner function \(f\).
In the function \(v(x) = \ln(9x)\), the chain rule comes into play. Here, the inner function is \(9x\) and the outer is \(\ln(x)\). Applying the chain rule, we first differentiate \(\ln(9x)\) with respect to \(9x\), giving \(\frac{1}{9x}\).
Then, multiply by the derivative of the inner function \(9x\), which is 9. This results in the simplified form \(v'(x) = \frac{1}{x}\). The chain rule simplifies otherwise complex derivatives into a straightforward multiplication process.

Differentiation Steps

To differentiate a function systematically, there are several steps you can employ. This structured approach ensures you catch every detail in the process, especially in complex functions:

• **Identify the function type**: Determine whether the function requires the product, chain, or quotient rule.
• **Apply relevant rules**: For our function \(f(x) = x^2 \ln(9x)\), we used the product and chain rules.
• **Differentiate**: Take the derivative of each component—both the product components and any nested functions.
• **Substitute back**: Place the derivatives into their appropriate spots according to rules used.
• **Simplify**: Finally, tidy up your results into the simplest form possible. For our function, it resulted in \(f'(x) = 2x \ln(9x) + x\).

Proceeding with logical steps not only improves accuracy but also enhances your understanding of how each piece affects the whole. By going through these differentiation steps systematically, one can tackle even the most daunting problems with confidence.