Differentiation of algebraic and trigonometric expressions can be used for calculating rates of change, stationary points and their nature, or the gradient and equation of a tangent to a curve.

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Differentiation is used in maths for calculating **rates of change**.

For example in mechanics, the rate of change of displacement (with respect to time) is the velocity. The rate of change of velocity (with respect to time) is the acceleration.

The rate of change of a function with respect to can be found by finding the derived function .

For an equation beginning , the rate of change can be found by differentiating with respect to . In its notation form this is written as . This is also known as 'Leibniz Notation'.

There are many ways a question can ask you to differentiate:

**Differentiate**the function...- Find
- Find
- Calculate the
**rate of change**of... - Find the
**derivative**of… - Calculate the
**gradient of the tangent**to the curve...

The general rule for differentiation is:

- In other words, you bring the power down to the front to multiply and subtract 1 from the power.

- Question
Differentiate

- Question
Find the derivative of

When calculating the rate of change or the gradient of a tangent to a curve, we are required to write the final answer to the differentiated expression **without negative or fractional powers**. Doing so makes it much easier to evaluate for specific values without a calculator.

To remove negative and fractional powers, we need to recall the laws of indices. The two that will be useful here are:

Find the rate of change of at .

Using , we find that:

This is very difficult to evaluate when without a calculator, so we need to use our laws of indices to change this into a positive power.

Now when ,

- Question
Find the gradient of the tangent to the curve with equation at the point when .

Now when ,

Therefore the gradient of the tangent to the curve is 1.

The previous examples have very simple expressions. Sometimes we aren't able to differentiate all expressions in their current form as we require the expression to be sums and/or differences of terms of the form .

Before differentiating:

- Remove brackets
- Separate 'top heavy' fractions
- Change terms involving roots into fractional powers
- Change terms with on the denominator to negative powers

Differentiate

Now we have it in the correct form we can differentiate.

- Question
Find the derivative of

We need to separate the top heavy fraction first then use our laws of indices to get into the correct form to be able to differentiate.

- Question
Find the derivative of

Remove the brackets first, then differentiate.