# What are the derivatives of trig functions

DERIVATIVES OF INVERSE TRIG FUNCTIONS . (top )

Again, these derivatives are true only when the angle x is expressed in radians. We'll follow the same general strategy for calculating each of these derivatives. First, we'll rewrite the function to remove the inverse expression. We'll then differentiate implicitly, and we'll finish off by using trig to rewrite all of each derivative in terms of x.

1. Derivative of Arcsine:

Explanation Let

. We can get rid of the inverse trig function by rewriting this as

.

Next, differentiate implicitly:

We must now replace "cos y " with some term involving x.

Since

, the triangle at left is formed. The bottom leg is found using the Pythagorean theorem. Using this triangle, we can see that

.

Substituting this into the equation for

, we find that

.

2. Derivative of Arccosine:

Explanation Let

. We can get rid of the inverse trig function by rewriting this as

.

Next, differentiate implicitly:

We must now replace "cos y " with some term involving x.

Since

, the triangle at left is formed. The bottom leg is found using the Pythagorean theorem. Using this triangle, we can see that

.

Substituting this into the equation for

, we find that

.

3. Derivative of Arctangent:

Explanation Let

. We can get rid

of the inverse trig function by rewriting this as

.

Next, differentiate implicitly:

We must now replace "cos y " with some term involving x.

Since

, the triangle at left is formed. The hypotenuse is found using the Pythagorean theorem. Using this triangle, we can see that

.

Substituting this into the equation for

, we find that

.

Of course, each of the reciprocal trig functions—cosecant, secant, and cotangent—also has a corresponding inverse function. Here, we evaluate the derivatives of arccosecant, arcsecant, and arccotangent, using the same methods.

4. Derivative of Arccosecant:

Explanation expl

5. Derivative of Arcsecant:

Explanation Let

. We can get rid of the inverse trig function by rewriting this as

.

Next, differentiate implicitly:

We must now replace "(sec y )(tan y )" with some term involving x.

Since

, the triangle at left is formed. The leg on the right side is found using the Pythagorean theorem. Using this triangle, we can see that

.

Substituting this into the equation for

, we find that

.

6. Derivative of Arccotangent:

Explanation expl

One final comment: If reciprocal ratios are inputs of inverse functions where the functions have reciprocal relationships, the resulting anges will be the same!

This fact can be used to rewrite problems and make them easier to solve.

Source: www.math.brown.edu

Category: Bank