Differentiating sin(x), cos(x) and tan(x)
When differentiating trigonometric functions we obtain the following derivatives:
Note that we can use instead of for particularly small . This is because for tiny , would be negligible. If included in the expansion, as with any other higher order terms, it disappears as and the same result holds.
where is a real constant. See Example 1. The derivative of is given in the Edexcel Formula Booklet along with the derivatives of the reciprocal functions as below. However, the rest of the derivatives seen on this page are not given and should be memorised. A question may ask you to differentiate linear combinations of trigonometric functions (see Example 2) or possibly even more complex functions (see product, quotient and chain rule).
Differentiating Reciprocal Trigonometric Functions
The derivatives of the reciprocal trigonometric functions are as follows:
Similarly to the above, we can also show using the chain rule that:
An exam question could ask all the usual things that require differentiation without explicitly stating it. See Example 2 for the use of differentiating reciprocal trigonometric functions in an example.
Differentiating Inverse Trigonometric Functions
The derivatives of the inverse trigonometric functions are as follows:
We can show these derivatives using the fact that (see more on this). Consider the derivative of , for example. We can write this as and differentiate with respect to : . Hence, it follows that . To write this as a function of , we note that . Recall that is one-to-one on the default interval . It follows that is positive on this interval. Hence . A similar process can be applied to the derivative of . See Example 3 for an example finding and using the derivative of .