Derivative of the Logarithmic Function ( y x^3 3 ln(x) - 1 )

In this article, we will explore the process of finding the derivative of the function ( y x^3 3 ln(x) - 1 ) using fundamental differentiation rules such as the product rule and the chain rule. We will break down the steps involved in the differentiation process and simplify the results to arrive at the final derivative.

Simplifying the Function Expression

The given function is:

$$ y x^3 3 ln(x) - 1 $$

Notice that the term (3 ln(x)) can be simplified to (3 ln(x) - 1) as a whole. This simplification helps us in applying the appropriate differentiation rules effectively.

Step 1: Identifying Components

To use the product rule, it is helpful to split the function into two components:

$$ u x^3 $$ and $$ v 3 ln(x) - 1 $$

Step 2: Finding the Derivatives

We will now find the derivatives of $$ u $$ and $$ v $$. Using the power rule for $$ u $$ and the chain rule for $$ v $$, we get:

$$ frac{du}{dx} frac{d}{dx} x^3 3x^2 $$

$$ frac{dv}{dx} frac{d}{dx} (3 ln(x) - 1) 3 cdot frac{1}{x} frac{3}{x} $$

Step 3: Applying the Product Rule

According to the product rule:

$$ frac{d}{dx} (uv) u frac{dv}{dx} v frac{du}{dx} $$

Substituting $$ u $$ and $$ v $$ with their derivatives, we have:

$$ frac{dy}{dx} x^3 cdot frac{3}{x} (3 ln(x) - 1) cdot 3x^2 $$

Step 4: Simplification

Let's simplify the expression step by step:

$$ frac{dy}{dx} x^3 cdot frac{3}{x} (3 ln(x) - 1) cdot 3x^2 $$

First term: $$ x^3 cdot frac{3}{x} frac{3x^3}{x} 3x^2 $$

Second term: $$ (3 ln(x) - 1) cdot 3x^2 9x^2 ln(x) - 3x^2 $$

Combining both terms:

$$ frac{dy}{dx} 3x^2 9x^2 ln(x) - 3x^2 $$

Further simplification:

$$ frac{dy}{dx} 9x^2 ln(x) $$

Conclusion

Thus, the derivative of the function ( y x^3 3 ln(x) - 1 ) is:

$$ boxed{9x^2 ln(x)} $$

This result demonstrates the effective use of fundamental differentiation techniques to simplify and find the derivative of a logarithmic function.