Derivative of `x*ln(x)`

Step-by-step differentiation

Answer

\frac{d}{d x} [x \cdot ln (x)]

ln (x) + 1

Step-by-step solution

1Product Rule
$\frac{d}{d x} [x \cdot ln (x)] = \frac{d}{d x} [x] \cdot ln (x) + x \cdot \frac{d}{d x} [ln (x)]$
Let $f = x$ and $g = ln (x)$ . The product rule says $(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}$ — find each factor's derivative, then assemble.
↳Find f′: Differentiate the first factor, $f = x$ :
2Power Rule
$\frac{d}{d x} [x] = 1$
$x$ is $x^{1}$ : by the power rule, the exponent $1$ comes down and the power drops to $x^{0} = 1$ , so the derivative is $1$ .
↳Find g′: Differentiate the second factor, $g = ln (x)$ :
3Derivative of ln
$\frac{d}{d x} [ln (x)] = \frac{1}{x}$
The derivative of $ln (x)$ is $\frac{1}{x}$ — a fundamental logarithm rule.
=Combine
$\frac{d}{d x} [x \cdot ln (x)] = 1 \cdot ln (x) + x \cdot \frac{1}{x} = ln (x) + 1$
With $f^{'} = 1$ and $g^{'} = \frac{1}{x}$ , substitute into $f^{'} \cdot g + f \cdot g^{'}$ and simplify.
✓Final Answer
$\frac{d}{d x} [x \cdot ln (x)] = ln (x) + 1$
The first derivative simplifies to $ln (x) + 1$ .

Understanding this derivative

Differentiating x·ln(x) exercises 3 distinct techniques — the product rule, the power rule and the natural log rule. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

Because two expressions are multiplied together, the product rule applies: differentiate each factor in turn while keeping the other fixed, then add the results. Note that the derivative of a product is not simply the product of the derivatives. The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. The natural logarithm differentiates to the reciprocal: d/dx[ln x] = 1/x. When the argument of the log is itself a function, the chain rule divides its derivative by that argument.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

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Derivative of x*ln(x)

Step-by-step solution

Understanding this derivative

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Derivative of `x*ln(x)`