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

Question

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)

View Explanation

Log in for full answers

Similar Questions

Differentiate 5 sin ⁡ ( 𝑥 3 + 2 𝑥 )

Which rule would be used to differentiate the function

Find dy/dt. y = cos()

If wanting to differentiate: 𝑦 = 1 𝑥 2 + 1 The most appropriate rule to use would be the [ Select ] chain general right handed product rule

If f ( x ) = 5 x − 1 , g ( 0 ) = 2 , g ′ ( 0 ) = 4 , and h ( x ) = f ( g ( x ) ) {"version":"1.1","math":"f(x)=5x-1,\ g(0)=2,\ g'(0)=4, \text{ and } h(x)=f(g(x))"}then what is the value of h ′ ( 0 ) ? {"version":"1.1","math":"h'(0)?"}

If wanting to differentiate: The most appropriate rule to use would be the chain rule

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler