Question

Find \(f^{\prime}(x)\) for each function. $$ f(x)=\ln \left(4 x^{3}+x\right) $$

Short answer

The derivative is \(f'(x) = \frac{12x^2 + 1}{4x^3 + x}\).

Step-by-step solution

Identify the Outer and Inner Functions

To find the derivative of the function \(f(x) = \ln(4x^3 + x)\), first note that this is a composite function. The outer function is \(\ln(u)\) where \(u = 4x^3 + x\), and the inner function is \(4x^3 + x\).

Differentiate the Outer Function

The derivative of \(\ln(u)\) with respect to \(u\) is \(\frac{1}{u}\). So, the derivative of the outer function \(\ln(4x^3 + x)\) with respect to \(u\) is:\[\frac{d}{du}\left( \ln(u) \right) = \frac{1}{4x^3 + x}\].

Differentiate the Inner Function

Now, find the derivative of the inner function \(4x^3 + x\) with respect to \(x\):\[\frac{d}{dx}(4x^3 + x) = 12x^2 + 1\].

Use the Chain Rule

Using the chain rule, the derivative \(f'(x)\) is given by:\[f'(x) = \frac{d}{dx}[\ln(4x^3 + x)] = \frac{1}{4x^3 + x} \times (12x^2 + 1)\].

Simplify the Expression

Finally, simplify the expression obtained from the chain rule:\[f'(x) = \frac{12x^2 + 1}{4x^3 + x}\].

Key concepts

Chain Rule

The Chain Rule is a fundamental technique in calculus for finding the derivative of composite functions. It helps us differentiate functions where one function is nested inside another. Think of peeling layers of an onion, except here, you're differentiating them. If you have a composite function like \(f(x) = \ln(4x^3 + x)\), the outer layer is \(\ln(u)\), and the inner layer is \(u = 4x^3 + x\). When using the Chain Rule, the process is to differentiate the outer function first and then multiply by the derivative of the inner function. Simply put, if you have \(y = g(h(x))\), where \(y\) is a function of \(h\), and \(h\) is a function of \(x\), then the derivative \(y'\) with respect to \(x\) is:

• First, differentiate the outer function: \(g'(h(x))\).
• Next, multiply by the derivative of the inner function: \(h'(x)\).

In our example, applying the chain rule means we take the derivative of \(\ln(u)\) to get \(\frac{1}{u}\), then multiply it by the derivative of \(4x^3 + x\). This results in the complete derivative of the function.

Logarithmic Differentiation

Logarithmic differentiation is especially helpful when dealing with functions involving logarithms, like those with the natural log, \(\ln\). It simplifies the process of differentiating complex functions. In cases where directly applying the chain rule or other differentiation techniques becomes cumbersome, logarithmic differentiation comes to the rescue. In our exercise, we used the derivative of a natural log while applying the chain rule. The function \(f(x) = \ln(4x^3 + x)\) was our target, and the derivative of \(\ln(u)\) is \(\frac{1}{u}\). Combining this with the chain rule enhances our ability to derive the correct derivative. Make sure to:

• Identify the complex function containing the logarithm.
• Apply the appropriate rules of derivatives for logarithmic functions.

Using this technique is handy for breaking down and simplifying even the trickiest functions.

Composite Functions

Composite functions are like nesting dolls in mathematics where one function is inside another. They appear when one function is applied, and the result is put into another function. Consider \(f(x) = \ln(4x^3 + x)\); here, the expression \(4x^3 + x\) is plugged into \(\ln(u)\), making it composite. The secret to mastering composite functions is understanding and identifying their layers. Recognize the stages:

• Outer function (\(\ln(u)\) in the example)
• Inner function (\(4x^3 + x\) in the example)

With this understanding, you can easily apply rules like the chain rule to differentiate them, creating a path that leads from the inside out.

Differentiation Techniques

Differentiation techniques are the sockets in your math toolbox. They allow you to tackle a variety of functions systematically, ensuring you get the right derivative efficiently. These techniques include rules like the Product Rule, Quotient Rule, and Chain Rule, which provide frameworks for attacking different types of function combinations.In our example function \(f(x) = \ln(4x^3 + x)\), using the Chain Rule was appropriate because it was a composite function. Each differentiation scenario may require a different rule:

• For products of two functions, use the Product Rule.
• For quotients, use the Quotient Rule.
• For chains (nested functions), use the Chain Rule.

By learning when to apply each technique, you can make differentiating any complex function manageable and less intimidating.