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Differentiation

Variable power functions aka Logarithmic Differentiation

You need to have mastered the Chain Rule before you start logarithmic differentiation.

When a function contains another function as a power, so the power is variable, we cannot use our previous methods.

Simple examples include 5x, 2x−3x and (x2 + 3x − 24)3 − 2x

The problem is easily solved if we first apply the natural log function to both sides and use the rules we learnt before.

Remember:   Hit it with a log! hit variable powers with the log


Example

Differentiate y = 8x

y equals f of x equals 8 to the power of x
Take natural logarithm of both sides then ln of f of x equals x times ln 8
Differentiate both sides. f dash of x over f of x equals ln 8
Multiply both sides by f of x. f dash of x equals (ln 8) times (f of x)
Substitute for f of x.  f dash of x equals (8 to the power of x) times (ln 8)

Now a more complicated expression. Differentiate y = f(x) = x2x.

y equals f of x equals x to the power of 2x
Take natural logarithm of both sides then ln of f of x equals 2x times ln x
Differentiate both sides, using chain rule on the left, product rule on the right. f dash of x over f of x equals 2 plus 2 times ln x
Multiply both sides by f of x. f dash of x equals (2 plus 2 times ln x) times (f of x)
Substitute for f of x.  f dash of x equals (x to the power of 2x) times (2 plus 2 times ln x)

The method is summarised:

  • take natural logarithms
  • differentiate (using the Chain Rule on the left side)
  • multiply by f(x)
  • substitute for f(x)

More examples (this will open a new window)

Exercise

First some simpler exercises to get you used to the method.

Differentiate the expression with respect to x.

 

f(x) =

 

Put your answer here:

Exercise

Now try your hand at some more challenging exercises.

Differentiate the expression.

 

f(x) =

 

Put your answer here:

Exercise

The previous exercises have a more limited scope than this exercise. Differentiate these variable power functions:

  1. y = f(x) = 2x
  2. y = f(x) = 55x
  3. y = f(x) = (x2 + 1)x
  4. y = f(x) = 10x
  5. y = f(x) = 3x(x2 + 1)                  Check first 5 answers (this will open a new window)
  6. y = f(x) = 4x
  7. y = f(x) = 7x − 3
  8. y = f(x) = (x2 − 1)x + 1
  9. y = f(x) = (10 − 2x)x
  10. y = f(x) = 10x(5x2)                  Check next 5 answers (this will open a new window)
  11. y = f(x) = (3x + 2)x
  12. y = f(x) = x4x
  13. y = f(x) = (x2 + 2)−3x
  14. y = f(x) = 10x3 − x
  15. y = f(x) = xx                  Check next 5 answers (this will open a new window)
  16. y = f(x) = 3x + 5
  17. y = f(x) = (2x)5x
  18. y = f(x) = (cos x )x
  19. y = f(x) = ex5x
  20. y = f(x) = 3x + 122x + 2                  Check last 5 answers (this will open a new window)

Differentiation index

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