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## 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! ### Example

Differentiate y = 8x     Now a more complicated expression. Differentiate y = f(x) = x2x.     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) =

### Exercise

Now try your hand at some more challenging exercises.

Differentiate the expression.

 f(x) =

### 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