Derivative of y = x^(ln x) using logarithms Step by Step

Step by step

The question

Question: find dy/dx for y = x^(ln x), using logarithms.

y = x^(ln x)

x is in the base and in the power.

The power rule needs a fixed power.

The bˣ rule needs a fixed base.

Neither works, so we take ln.

x is in the base AND the power

Careful!

The power rule is only for a fixed power.

Here the power changes with x.

Careful: not x^(ln x) → (ln x)x^(ln x − 1) ✗

1 Take ln of both sides

Take ln of both sides.

Both sides stay equal.

ln y = ln(x^(ln x))

2 Use the log rules

The log rule: a power inside ln comes down in front.

ln(a^b) = b ln a

So ln y is ln x, times ln x.

No more power!

ln y = (ln x) ln x

3 The derivative of each side

The left side: y depends on x, so the chain rule gives 1/y, times dy/dx.

(ln y)′ = 1y × dydx

The right side is a product: use the product rule.

ln x, gives 1/x, and ln x gives 1 over x.

1ydydx = 1x × ln x + (ln x) × 1x

4 Multiply by y

Multiply both sides by y, to get dy/dx alone.

dydx = y(1x × ln x + (ln x) × 1x)

5 Put y back

Put y = x^(ln x) back, and add the two equal parts: x^(ln x) × 2 ln x over x.

dydx = x^(ln x) × 2 ln xx

6 Check

Let's check: the derivative is the slope.

At x = 2, our answer gives 1.1207, with a calculator.

dy/dx at x = 2: 1.1207

Now the slope between two very close points: x = 2 and x = 2.001.

A calculator gives about 1.12.

Very close to 1.1207.

Right!

y(2.001) − y(2)0.001 ≈ 1.12

Let's look back.

First: take ln of both sides.

Then: use the log rules.

Then: the derivative of each side.

Then: multiply by y.

Then: put y back.

Last: check.

Answer