Remember that the gradient of a straight line is *m*. The tangent to the line is the line itself, so the gradient of tangent is *m* everywhere along the line.

So if .

For a horizontal line in the form *y=k*, the gradient is zero.

So if .

In short, we could say that:

A more general way of writing the rule is:

OR

Also, if there are several terms in an expression, we differentiate each one separately. We can write this as a rule:

### Examples:

Differentiate each function:

1.

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2.

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3.

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4. If , evaluate

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5. Differentiate with respect to *r*.

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In this case, we are differentiating with *r* as the variable, so and *h* are constants.

## Composite Function Rule

A **composite**** function** is a function composed of two or more other functions. For example, is made up of a function where .

To differentiate a composite function, we need to use the result:

### Examples

1. Differentiate

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2. Differentiate

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The derivative of a composite function is the product of two derivatives. One is the derivative of the function inside the brackets. The other is the derivative of the whole function.

### Examples:

Differentiate:

1.

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2.

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## Product and Quotient Rule

Differentiating the product of two functions gives the result:

If

Differentiating the quotient of two function gives the result:

If

### Examples

1. Differentiate

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)

2. Differentiate

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3. Differentiate

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4. Differentiate

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