Mathematics·Derivative·grades 10–11

Find the derivative of the function f(x) = x·ln(x)

Problem

The function f(x) = x·ln(x) is given. Find its derivative f′(x).

Answer

f′(x) = ln(x) + 1

Step-by-step solution

Step 1. This is a product
The function is a product of two factors: $u = x$ and $v = \ln x$ . The rule: $(uv)' = u'v + uv'$ .
Step 2. Differentiate each factor
$u' = (x)' = 1$ , and $v' = (\ln x)' = \dfrac{1}{x}$ .

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Step 3. Substitute into the formula
Step 4. Answer

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Common mistakes

Differentiating factor by factor as $u'\cdot v'$ — that is wrong, the product rule is required.
Not cancelling $x\cdot\frac1x = 1$ and leaving a clumsy answer.

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