**1.** We should recall the chain rule applied to gives us

(1)

where . So what? Well, if is some nightmarish function (e.g. ), then we have

(2).

Quite a quick way to compute nightmarish derivatives!

**2. Example.** What is the derivative of ?

*Solution*: we should note that

(3).

This has quite an easy derivative:

(4)

Thus we find

(5).

**3. Example.** What is the derivative of ?

*Solution*: We see

(6).

This has quite an easy derivative:

(7)

Thus we find

(8).

This concludes our second example.

**4. Example.** What is the derviative of ?

*Solution*: We take its logarithm, which is

(9).

Now we take the derivative of the logarithm:

(10).

Then we multiply both sides by to get the derivative

(11).

**Exercise 1.** Differentiate .

**Exercise 2.** Differentiate .

**Exercise 3.** Differentiate .

**Exercise 4.** Differentiate .

**Exercise 5.** Differentiate .

**Exercise 6.** Differentiate .

**Exercise 7.** Differentiate the function .

**Exercise 8.** Differentiate .

**Acknowledgement:** these examples and exercises were borrowed and/or modified from Dr Kouba’s Calculus Page, although my solutions were computed by myself.

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