Differentiating sin(x) and cos(x)
In AS you will have only differentiated expressions with xn. Now in A2, you will be looking at more cases. Let's begin with sin(x) and cos(x).
The general formulae for differentation of sin(x) and cos(x) is:
You will need to know how to prove the derivative of expressions with sin(x) and cos(x) using differentiation from first principles. The principle is the same from AS, but you need to use some limits.
Here is a reminder of the small angle approximations for sinx and cosx:
These small angle approximations can be used to get to the limits you need to use for the proofs. You will not need to know this specific part for the exam, but it is good to set where these limits come from.
Example 1:
Note: Like the AS proofs, carefully look at each step, especially the step where you are using the limits to get to the derivative, as this shows your understanding on where you have to use those limits.
Now these are examples of differentiating expressions using the formulas without using first principles.
Example 2:
Practice Question:
Note: These differentiating using first principles examples show how to solve the problem using one solution. It may be useful to check out the markscheme for these types of questions, which will usually show many methods to solve the question, and see which method suites you. But then remember you must stick with this method to avoid confusion as exam boards are very specific with how they award marks for these types of questions.