Third Derivative of Ln- Calculus Explanation

What Is the Third Derivative of ln(x)?

Here's the answer, straight up: the third derivative of ln(x) is 2/x³. No buildup, no suspense needed.

If you want to understand why it's 2/x³ and how to derive it yourself, keep reading. If you just needed the answer, there it is.

The Step-by-Step Derivation

Derivatives build on each other. You can't skip to the third derivative without doing the first two. Here's how it works:

Step 1: First Derivative

f(x) = ln(x)

Apply the basic derivative rule for natural log:

f'(x) = 1/x

This is the part you should already know. If you don't, go back and memorize it before moving on.

Step 2: Second Derivative

Now differentiate f'(x) = 1/x.

Rewrite it as x^(-1) to make the power rule easier to apply:

f''(x) = d/dx [x^(-1)] = -1 · x^(-2) = -1/x²

Step 3: Third Derivative

Differentiate f''(x) = -x^(-2):

f'''(x) = d/dx [-x^(-2)] = -(-2) · x^(-3) = 2x^(-3) = 2/x³

Done. That's the third derivative.

The Pattern You're Actually Looking For

Once you see the pattern, higher-order derivatives of ln(x) become predictable:

• 1st derivative: 1/x = 1 · x^(-1)
• 2nd derivative: -1/x² = -1 · 2 · x^(-2)
• 3rd derivative: 2/x³ = 1 · 2 · 6 · x^(-3)

Notice the factorial pattern? The coefficients are 1!, then 1! · 2!, then 1! · 2! · 3!. The sign alternates starting from the second derivative.

The General Pattern

The nth derivative of ln(x) follows this formula:

f⁽ⁿ⁾(x) = (-1)ⁿ⁻¹ · (n-1)! / xⁿ

For n ≥ 1. Plug in n = 3 and you get:

(-1)³⁻¹ · (3-1)! / x³ = (-1)² · 2! / x³ = 1 · 2 / x³ = 2/x³

Matches what we got above.

Quick Reference Table

The pattern holds: coefficient is (n-1)! with alternating sign starting negative at the 2nd derivative.

Common Mistakes to Avoid

• Forgetting the chain rule on rewrites — when you rewrite 1/x as x^(-1), the power rule applies directly. Don't multiply by extra factors.
• Losing the negative sign — signs flip with each differentiation. Check your work if you end up with the wrong sign.
• Wrong factorial — the third derivative has 2! = 2 in the numerator, not 3! = 6. That's the 4th derivative.

How to Check Your Work

When you compute derivatives of ln(x) and want to verify:

1. Confirm the first derivative is definitely 1/x
2. Apply the power rule consistently: d/dx[x^n] = n·x^(n-1)
3. Track the sign through each step
4. Verify the coefficient matches (n-1)! for the nth derivative

If any step breaks down, go back. The math is deterministic here — there's no ambiguity in the answer.

Why This Matters

You won't find yourself computing the third derivative of ln(x) in everyday applications. But understanding the pattern matters for:

• Taylor series expansions of log functions
• Higher-order calculus problems where patterns emerge
• Verifying your derivative skills

If you're working through calculus coursework, this is a skill that shows up on exams. Know it.

The third derivative of ln(x) is 2/x³. That's the answer. No elaboration needed beyond this point.