The

*basic trigonometric functions* include the following 6 functions:

*sine* (sin *x*),

*cosine* (cos *x*),

*tangent* (tan *x*),

*cotangent* (cot *x*),

*secant* (sec *x*) and

*cosecant* (csc *x*).

For each of these functions, there is an

inverse trigonometric function.
The are called the

*inverse sine* (arcsin *x*),

*inverse cosine* (arccos *x*),

*inverse tangent* (arctan *x*),

*inverse cotangent* (arccot *x*),

*inverse secant* (arcsec *x*) and

*inverse cosecant* (arccsc *x*), respectively.

All these functions are continuous and differentiable in their domains. Below we make a list of derivatives for these 12 functions.

Derivatives of Basic Trigonometric Functions

We have already derived the derivatives of sine and cosine on the

Definition of the Derivative page.
They are as follows:

Using the

quotient rule
it is easy to obtain an expression for the derivative of

*tangent*:

The derivative of

*cotangent* can be found in the same way. However, this can be also done using
the

chain rule for differentiating
a composite function:

Similarly, we find the derivatives of

*secant* and

*cosecant*:

Derivatives of Inverse Trigonometric Functions

The derivatives of the inverse trigonometric functions can be derived using the

inverse function theorem.
For example, the sine function

*x = φ* (*y*) = sin *y*
is the inverse function for

*y = f* (*x*) = arcsin *x*.
Then the derivative of

*y* = arcsin *x* is given by

Using this technique, we can find the derivatives of the other inverse trigonometric functions:

In the last formula, the absolute value |

*x*| in the denominator appears due to the fact that the product

tan *y* sec *y* should always be positive in the
range of admissible values of

*y*, where

*y* ∈ (0, *π*/2) ∪ (*π*/2, *π*),
i.e. the derivative of the

*inverse secant* is always positive.

Similarly, we can obtain an expression for the derivative of the

*inverse cosecant* function:

Table of Derivatives of Trigonometric Functions

The derivatives of 6 basic trigonometric functions and 6 inverse trigonometric functions considered above are presented
in

the following table:

In the examples below, find the derivative of the given trigonometric function.