Revision:Differentiation of Trigonometry Functions

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What you need to know

It is possible to find the derivative of trigonometric functions. Here is a list of the standard forms that you need to know:

One condition upon these results is that x must be measured in radians.

Proof

To differentiate sin x from first principles, let us consider the gradient m of the line between the points (x, sin x) and (x+o, sin(x+o)), where o is small.

As o tends to 0, [sin(o/2)]/(o/2) tends to 1, and cos(x + o/2) tends to cos x, and so m tends to cos x; that is,

Remember that cos x = sin(π/2 - x). Replacing x with (π/2 - x), and using the chain rule, in the above:

That is,

Using the quotient rule:

Deriving the final two by noting that cot x = tan(π/2 - x) and csc x = sec(π/2 - x) respectively:

Applying the Chain Rule: An example

The chain rule is used to differentiate harder trigonometric functions.

Differentiate with respect to x.

Let

Let

therefore

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