# Derivatives of inverse trigonometric functions

Inverse trigonometric functions are the inverse of trigonometric functions .

For example if, y = sinx then the inverse function of y = sinx is , is denoted by: x=sin^{-1}y and is called inverse sin function.

You should note that: doesn’t means instead

“y = sin ^{-1} x” is the inverse function of “x = sin y”

**Derivatives of Inverse Trigonometric Functions**:

The derivatives of inverse sine , inverse cos , inverse tan , inverse csc , inverse sec , inverse cot functions are given below:

**Derivative of inverse sin function**:

proof:

If , then the function is called inverse sin function.

If, then ,

now if we differentiate with respect to x , using implicit differentiation technique then,

Now using the trigonometric formula,

Now as , sin y = x

Thus ,

**Derivative of inverse cos function**:

proof:

If , then the function is called inverse cos function.

And, If, then we can also rewrite is as:

now if we differentiate with respect to x , using implicit differentiation technique then,

Now using the trigonometric formula,

Now as , cos y = x

Thus ,

**Derivative of inverse tan function**:

proof:

When , then the function “f” or y is called inverse tan function.

and we can also equally re-write above function as:

If we differentiate both L.H.S and R.H.S of the equation with respect

to “y”.

then,

Now using the concept of differentials we can re write above equation as:

Thus ,

**Derivative of inverse csc , inverse sec & inverse cot functions**:

We can use the similar method we used above to find derivative of inverse sin , cos and tan function to find the derivatives of inverse csc , sec and cot function.

After differentiation we get following result:

**Derivative of inverse csc function**:

**Derivative of inverse sec function**:

**Derivative of inverse cot function**:

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