Use the chain rule to find the derivative of the following function: f(x) = loge(x3)

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BlackTom AI · Accepted Answer

We start by identifying the function and the rule to apply. The function is f(x) = ln(x^3), and we will use the chain rule for derivatives of natural logarithms: d/dx [ln(u)] = u'/u, where u is a inner function of x.

Option a: f'(x) = (1/x) loge(x^3)(3x^2). This expression multiplies several factors in a way that doesn’t align with the standard chain rule result for ln(x^3). The derivative should be a single ratio of u' to u, not a product of log terms with powers of x; the structure here is inconsistent with d/dx[ln(u)].

Option b: f'(x) = loge(3x^2). This would imply replacing the derivative with a logarithm of the inner derivative, which is not how the chain rule works. The derivative should simplify to a rational expression, not a log of a function.

Option c: f'(x) = (1/x)loge(3x^2). Here we see a product of 1/x with ln(3x^2). The chain rule for ln(u) yields u'/u, which would correspond to a ratio, not a logarithm multiplied by 1/x. The presence of ln(3x^2) in the numerator indicates an incorrect application.

Option d: f'(x) = (1/x^3)(3x^2). This matches the chain rule correctly: with u = x^3, u' = 3x^2, and derivative is u'/u = (3x^2)/(x^3) = (1/x^3)(3x^2). This expression also simplifies to 3/x, which is the correct derivative for ln(x^3).

In summary, only option d adheres to the chain rule for the natural logarithm of a composite inside function, while the other options misapply the rule or misplace operations.

Use the chain rule to find the derivative of the following function: f(x) = loge(x3)

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