Question

Evaluate the following derivatives. $$\frac{d}{d x}\left(\ln ^{3}\left(3 x^{2}+2\right)\right)$$

Short answer

Question: Find the derivative of the function $y=\ln^3(3x^2+2)$ with respect to $x$. Solution: The derivative of the function is given by $\frac{dy}{dx}=\frac{18x\left(\ln\left(3x^2+2\right)\right)^2}{3x^2+2}$.

Step-by-step solution

Identify the outer and inner functions

We first identify the outer function and the inner function in our expression. The outer function is \(u^3\) (where \(u=\ln\left(3x^2+2\right)\)) and the inner function is \(v=\ln \left(3x^2+2\right)\). We will apply the Chain Rule using these functions.

Apply the Chain Rule to find the derivative of the outer function

Now that we have identified the outer and inner functions, we can apply the Chain Rule: $$\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}$$ We first find the derivative of the outer function with respect to \(u\) (where \(y=u^3\)): $$\frac{dy}{du} = 3u^2$$

Find the derivative of the inner function

Next, we need to find the derivative of the inner function with respect to \(v\) (where \(u=\ln v\)): $$\frac{du}{dv}=\frac{1}{v}$$

Find the derivative of v(x) with respect to x

Finally, we need to find the derivative of \(v\) with respect to \(x\) (where \(v=3x^2+2\)): $$\frac{dv}{dx}=6x$$

Apply the Chain Rule and simplify

Now, we plug in all the derivatives we found into the Chain Rule and simplify: $$\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}=3u^2 \cdot \frac{1}{v} \cdot 6x$$ Now, we substitute the original expressions of \(u\) and \(v\) back in: $$\frac{dy}{dx}=3\left(\ln\left(3x^2+2\right)\right)^2 \cdot \frac{1}{3x^2+2} \cdot 6x$$ Finally, we simplify the expression: $$\frac{dy}{dx}=\frac{18x\left(\ln\left(3x^2+2\right)\right)^2}{3x^2+2}$$

Key concepts

chain rule

The Chain Rule is a fundamental tool in calculus that is used to find the derivative of composite functions. A composite function is simply a function that is made up of two or more functions. For example, in our problem, we have the function \(\ln^3(3x^2 + 2)\), which combines the logarithmic function and a polynomial function.
The basic idea of the Chain Rule is to differentiate the outer function first and then multiply by the derivative of the inner function. This is summarized in the formula:

• \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

In this exercise, the outer function is \(u^3\) and the inner function is \(\ln(3x^2 + 2)\). By applying the Chain Rule, we first find the derivative of the outer function \(u\) with respect to \(x\). Then, we multiply it by the derivative of the inner function. This approach allows us to tackle complex derivatives step by step.

logarithmic differentiation

Logarithmic differentiation is another powerful technique in differentiation. It is especially useful when dealing with functions where one component is a logarithmic function, or when the function itself is a logarithm. In this problem, it helps us manage the derivative of \(\ln^3(3x^2+2)\) by breaking it down.

• When we differentiate a logarithm, we use the derivative rule \( \frac{d}{dx} [\ln(f(x))] = \frac{f'(x)}{f(x)} \).

For our specific problem, we start by differentiating the outer logarithmic function which is raised to a power. We essentially treat the power as part of the outer function to make use of the Chain Rule effectively.
Logarithmic differentiation simplifies the differentiation process by using natural log properties to turn complex multiplication and exponential functions into simpler ones for differentiation.

derivative of logarithmic function

The derivative of a logarithmic function is a crucial part of calculus, especially when dealing with problems involving growth or decay. The key property is that the derivative of \(\ln(x)\) is \(\frac{1}{x}\). This gives us a simple but powerful tool for finding rates of change in logarithmic terms.
In our exercise, the relevant logarithmic function is \(\ln(3x^2 + 2)\). To find its derivative, we apply the rule for the derivative of a natural log:

• \(\frac{d}{dx}[\ln(3x^2+2)] = \frac{1}{3x^2+2} \cdot (3x^2+2)' \)
• Realizing that \((3x^2+2)' = 6x\), the full derivative becomes \(\frac{6x}{3x^2+2}\).

This step is integral to the Chain Rule process. It gives us the rate of change of the inner function, which we then combine with the rate of the outer function to get the final derivative. Recognizing and applying the rule for logarithmic derivatives simplifies calculations and allows one to handle complex logarithmic expressions more rigorously.