Final answer:
To differentiate the function h(x) = ln(x * x² - 3), we apply the chain rule by differentiating the outside function (natural log) and multiplying it by the derivative of the inside function (x³ - 3), resulting in h'(x) = 3x² / (x³ - 3).
Step-by-step explanation:
The question asks us to differentiate the function h(x) = ln(x * x² - 3). To solve this, let us revisit a fundamental rule of logarithms that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. In differentiating logarithmic functions, we use the chain rule, which allows us to differentiate compositions of functions.
First, we rewrite the function for clarification: h(x) = ln(x³ - 3). The derivative h'(x) can be found by applying the chain rule:
- Differentiate the function, keeping the inside function as it is. The derivative of ln(u) with respect to u is 1/u.
- Multiply by the derivative of the inside function, which in this case is 3x².
Combining these steps, we get:
h'(x) = 1/(x³ - 3) × 3x²
simplifying to:
h'(x) = 3x² / (x³ - 3)