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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 5^x, 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 = 8^x

Now a more complicated expression. Differentiate y = f(x) = x^2x.

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.

Exercise

Now try your hand at some more challenging exercises.

Exercise

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

1. y = f(x) = 2^x
2. y = f(x) = 5^5x
3. y = f(x) = (x² + 1)^x
4. y = f(x) = 10^x
5. y = f(x) = 3^x(x² + 1)                  Check first 5 answers (this will open a new window)
6. y = f(x) = 4^x
7. y = f(x) = 7^{x − 3}
8. y = f(x) = (x² − 1)^{x + 1}
9. y = f(x) = (10 − 2x)^x
10. y = f(x) = 10^x(5x²)                  Check next 5 answers (this will open a new window)
11. y = f(x) = (3x + 2)^x
12. y = f(x) = x^4x
13. y = f(x) = (x² + 2)^−3x
14. y = f(x) = 10x^{3 − x}
15. y = f(x) = x^x                  Check next 5 answers (this will open a new window)
16. y = f(x) = 3^{x + 5}
17. y = f(x) = (2x)^5x
18. y = f(x) = (cos x )^x
19. y = f(x) = e^x5^x
20. y = f(x) = 3^{x + 1}2^{2x + 2}                  Check last 5 answers (this will open a new window)

Differentiation index