Derivative of x*ln(x)

Answer

\ln\left(x\right)+1

Derivative of x*ln(x)

Rule Used: Product Rule
Steps: Identify a product of two functions of x in x* $\log(x)$ : use the product rule (uv)' = u'v + uv'. Each factor is differentiated once while the other is held as-is, then the two terms are added.
Differentiate each factor and combine.
Final answer: 1+ $\ln(x)$

d/dx x ln(x) = 1+ln(x). The expression needs x > 0. At 0 and on the negatives it is undefined as a real function, so the derivative is too.

Instant step-by-step solution for x*ln(x)

Derivative LogarithmicIntermediate

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Derivative of x*ln(x) = 1+ln(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-x-ln-x

Derivative

x ln(x) → 1+ln(x)engine: 1+ln(x)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

x ln(x) is a product. Differentiating ln(x) produces 1/x, and x·(1/x) collapses to 1. Differentiating the x and leaving ln(x) produces the other term, ln(x). The sum is ln(x) + 1. On this site log means the natural log, so the engine prints 1+log(x), which is the same expression. The bare 1 is the part students drop.

Apply the product rule to x ln(x). The form to remember, already simplified, is 1+log(x).

The expression needs x > 0. At 0 and on the negatives it is undefined as a real function, so the derivative is too.

At x = 1, substitute into the simplified form 1+log(x). That number is the slope of x ln(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = x ln(x).
Apply the product rule to x ln(x). The form to remember, already simplified, is 1+log(x).
At x = 1, substitute into the simplified form 1+log(x). That number is the slope of x ln(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering x*(1/x)

Stopping at x·(1/x) = 1 forgets the term from differentiating x. The answer is not the constant 1; ln(x) is still there.

Differentiating only one factor of x ln(x)

Both factors depend on x. Differentiating one and copying the other, then stopping, drops a whole term.

Product rule — The rule page lists this example and the others in the same family.
Logarithmic derivatives — The rule page lists this example and the others in the same family.
Derivative of x sin(x) — Same family of differentiation, different function.
Derivative of x cos(x) — Same family of differentiation, different function.
Derivative of x e^x — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of x ln(x)?

1+ln(x). x ln(x) is a product. Differentiating ln(x) produces 1/x, and x·(1/x) collapses to 1. Differentiating the x and leaving ln(x) produces the other term, ln(x). The sum is ln(x) + 1. On this site log means the natural log, so the engine prints 1+log(x), which is the same expression. The bare 1 is the part students drop.