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Logarithmic differentiation

Calculus 1 Β· 22 problems, easy to harder

β–Ά Watch the first one solved

How we do it

  1. Take ln of both sides
  2. Use the log rules
  3. The derivative of each side
  4. Multiply by y
  5. Put y back
  6. Check

The steps for Derivative of y = x^x using logarithms. Every solution names its steps, so you always know what comes next.

Worked examples

Derivative of y = x^x using logarithms Easy

The question

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

y = xΛ£

x is in the base and in the power.

The power rule needs a fixed power.

The bΛ£ 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 xΛ£ β†’ xx^(x βˆ’ 1) βœ—

1 Take ln of both sides

Take ln of both sides.

Both sides stay equal.

ln y = ln(xΛ£)

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 x times ln x.

No more power!

ln y = 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)β€² = 1y Γ— dydx

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

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

1ydydx = 1 Γ— ln x + x Γ— 1x

4 Multiply by y

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

dydx = y(1 Γ— ln x + x Γ— 1x)

5 Put y back

Put y = x^(x) back, and tidy the bracket: xΛ£(ln x + 1).

dydx = xΛ£(ln x + 1)

6 Check

Let's check: the derivative is the slope.

At x = 2, our answer gives 6.7726, with a calculator.

dy/dx at x = 2: 6.7726

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

A calculator gives about 6.78.

Very close to 6.7726.

Right!

y(2.001) βˆ’ y(2)0.001 β‰ˆ 6.78

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
dydx = xΛ£(ln x + 1)

β–Ά Watch it solved

Derivative of y = x^(sin x) using logarithms Medium

The question

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

y = x^(sin x)

x is in the base and in the power.

The power rule needs a fixed power.

The bΛ£ 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^(sin x) β†’ (sin x)x^(sin x βˆ’ 1) βœ—

1 Take ln of both sides

Take ln of both sides.

Both sides stay equal.

ln y = ln(x^(sin 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 sin x, times ln x.

No more power!

ln y = (sin 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)β€² = 1y Γ— dydx

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

sin x, gives cos x, and ln x gives 1 over x.

1ydydx = cos x Γ— ln x + (sin x) Γ— 1x

4 Multiply by y

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

dydx = y(cos x Γ— ln x + (sin x) Γ— 1x)

5 Put y back

Put y = x^(sin x) back, and tidy the bracket: x^(sin x) (cos x ln x + sin x/x).

dydx = x^(sin x)(cos x ln x + sin xx)

6 Check

Let's check: the derivative is the slope.

At x = 2, our answer gives 0.3121, with a calculator in radians.

dy/dx at x = 2: 0.3121

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

A calculator gives about 0.31.

Very close to 0.3121.

Right!

y(2.001) βˆ’ y(2)0.001 β‰ˆ 0.31

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
dydx = x^(sin x)(cos x ln x + sin xx)

β–Ά Watch it solved

Derivative of y = (x + 1)^3/(x + 2)^2 using logarithms Harder

The question

Question: find dy/dx for y = (x + 1)Β³ over (x + 2)Β², using logarithms.

y = (x + 1)3(x + 2)2

The log rules turn a product into a sum, a fraction into a difference, and bring powers down.

ln(ab) = ln a + ln b
lnab = ln a βˆ’ ln b
ln(a^n) = n ln a

Careful!

The ln of a product is a sum of lns, not a product.

Careful: not ln(ab) = ln a Γ— ln b βœ—

1 Take ln of both sides

Take ln of both sides.

ln y = ln(x + 1)3(x + 2)2

2 Use the log rules

Use the log rules: each power comes down, multiplying becomes adding, dividing becomes taking away.

ln y = 3 ln(x + 1) βˆ’ 2 ln(x + 2)

3 The derivative of each side

The left side: 1/y, times dy/dx, by the chain rule.

Each ln(x + c) gives 1/x + c.

(ln y)β€² = 1y Γ— dydx
(ln(x + c))β€² = 1x + c

Each power waits in front.

1ydydx = 3x + 1 βˆ’ 2x + 2

4 Multiply by y

Multiply both sides by y.

dydx = y(3x + 1 βˆ’ 2x + 2)

5 Put y back

Put y back.

dy/dx = (x + 1)Β³ over (x + 2)Β² Γ— (3/(x + 1) βˆ’ 2/(x + 2)).

Books leave it like this.

dydx = (x + 1)3(x + 2)2(3x + 1 βˆ’ 2x + 2)

6 Check

Let's check: the derivative is the slope.

At x = 2, our answer gives 0.8438, with a calculator.

dy/dx at x = 2: 0.8438

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

A calculator gives about 0.84.

Very close to 0.8438.

Right!

y(2.001) βˆ’ y(2)0.001 β‰ˆ 0.84

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
dydx = (x + 1)3(x + 2)2(3x + 1 βˆ’ 2x + 2)

β–Ά Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. derivative of y = x^x using logarithms
    Answer
    dydx = xΛ£(ln x + 1)
  2. derivative of y = x^(2x) using logarithms
    Answer
    dydx = x^(2x)(2 ln x + 2)
  3. derivative of y = x^(3x) using logarithms
    Answer
    dydx = x^(3x)(3 ln x + 3)
  4. derivative of y = x^(5x) using logarithms
    Answer
    dydx = x^(5x)(5 ln x + 5)
  5. derivative of y = x^(x^2) using logarithms
    Answer
    dydx = x^(xΒ²)(2x ln x + x)
  6. derivative of y = x^(ln x) using logarithms
    Answer
    dydx = x^(ln x) Γ— 2 ln xx

Medium

  1. derivative of y = x^(sin x) using logarithms
    Answer
    dydx = x^(sin x)(cos x ln x + sin xx)
  2. derivative of y = x^(cos x) using logarithms
    Answer
    dydx = x^(cos x)(βˆ’sin x ln x + cos xx)
  3. derivative of y = (x + 1)^2 (x - 3)^4 using logarithms
    Answer
    dydx = (x + 1)2(x βˆ’ 3)4(2x + 1 + 4x βˆ’ 3)
  4. derivative of y = (x + 2)^3 (x - 1)^2 using logarithms
    Answer
    dydx = (x + 2)3(x βˆ’ 1)2(3x + 2 + 2x βˆ’ 1)
  5. derivative of y = (x + 3)^2 (x + 1) using logarithms
    Answer
    dydx = (x + 3)2(x + 1)(2x + 3 + 1x + 1)
  6. derivative of y = (x + 2)^4 (x - 3) using logarithms
    Answer
    dydx = (x + 2)4(x βˆ’ 3)(4x + 2 + 1x βˆ’ 3)
  7. derivative of y = (x + 1)^3 (x - 2)^2 using logarithms
    Answer
    dydx = (x + 1)3(x βˆ’ 2)2(3x + 1 + 2x βˆ’ 2)
  8. derivative of y = (x - 1)^4 (x + 5)^3 using logarithms
    Answer
    dydx = (x βˆ’ 1)4(x + 5)3(4x βˆ’ 1 + 3x + 5)

Harder

  1. derivative of y = (x + 1)^3/(x + 2)^2 using logarithms
    Answer
    dydx = (x + 1)3(x + 2)2(3x + 1 βˆ’ 2x + 2)
  2. derivative of y = (x - 2)^2/(x + 3) using logarithms
    Answer
    dydx = (x βˆ’ 2)2(x + 3)(2x βˆ’ 2 βˆ’ 1x + 3)
  3. derivative of y = (x + 4)^5/(x + 1)^3 using logarithms
    Answer
    dydx = (x + 4)5(x + 1)3(5x + 4 βˆ’ 3x + 1)
  4. derivative of y = (x + 5)^2/(x - 1)^4 using logarithms
    Answer
    dydx = (x + 5)2(x βˆ’ 1)4(2x + 5 βˆ’ 4x βˆ’ 1)
  5. derivative of y = (x - 3)^3/(x + 1)^2 using logarithms
    Answer
    dydx = (x βˆ’ 3)3(x + 1)2(3x βˆ’ 3 βˆ’ 2x + 1)
  6. derivative of y = (x + 1)(x + 2)(x + 3) using logarithms
    Answer
    dydx = (x + 1)(x + 2)(x + 3)(1x + 1 + 1x + 2 + 1x + 3)
  7. derivative of y = (x + 1)^2 (x + 3)^3/(x + 2) using logarithms
    Answer
    dydx = (x + 1)2(x + 3)3(x + 2)(2x + 1 + 3x + 3 βˆ’ 1x + 2)
  8. derivative of y = (x - 1)(x + 4)^2/(x + 2)^3 using logarithms
    Answer
    dydx = (x βˆ’ 1)(x + 4)2(x + 2)3(1x βˆ’ 1 + 2x + 4 βˆ’ 3x + 2)