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## Strategy for Mixed Problems

A very important skill in differentiating complex functions is recognising which methods in which order will compute the derivative.

We assume that you have mastered these already.

The key to successful differentiation is to determine what type of function you have.

• If the function contains only a basic function use the basic derivatives.
• If the function contains only a basic function multiplied by a constant use the Multiple Rule.
• If the function is a sum of terms use the Sum and Difference Rules.
• If the function is made up of two functions that are multiplied use the Product Rule.
• If the function is made up of two functions that are divided use the Quotient Rule.
• If the function is a composite function use the Chain Rule.
• If the function has a power that contains the variable use Logarithmic Differentiation.
• If you can see a way of simplifying the function you may use algebraic manipulation first. Examples include multiplying out a product or dividing a quotient to obtain simpler terms to differentiate or using exponent or logarithm properties to simplify a complicated exponent or logarithm expression.

### Examples

sin x is a basic function.
3 + sin x is a sum of terms.
sin (x + 3) is a composite function.
3sin x is a variable power function.
3sin x is a basic function multiplied by a constant.
(sin x)(cos x) is a product.
ln (esin x) can be simplified algebraically by recognising that ln and e are inverses so the expression simplifies to sin x.

### Exercises

For each function decide which method should be used first to obtain the derivative.

1.    y = sin (x3 + 2)

2.   y = x5 + 2x4 + 3x3 + 4x2 + 5x + 6

3.    y = x4(x3x2)

4.    y = ax − 2

5.    y = x2sin x

6.    y = 7.    y = xe2x − 7x

8.    y = 9.    y = 10.    y =  ln sin x3

For the next problems, which rules do you need to use in order to obtain the derivative?

1. y = 4ex(1 + x)(sin x)
2. y = 10−3x
3. y = ln (5x3 + 6)
4. y = 5. y = xe−x
6. y = cosec 2x
7. y = sin x(1 − cos2x)
8. y = ln (4 − e3x)
9. y = 3xe3 − x
10. y = x2 ln (4x2 − 1)                  Check all answers (this will open a new window)
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