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# Find the derivative of arcsin(sin x)

Find the derivative of the function

Using the chain rule and the formula for the derivative of arcsine we have

This is valid for all such that .

# Find the derivative of arccos (1/x)

Find the derivative of the function

By definition of arcsecant we know

We computed the derivative of in this exercise (Section 6.22, Exercise #4) so,

valid for .

# Find the derivative of arccos ((1-x)/21/2)

Find the derivative of the function

Using the formula for the derivative of and the chain rule we have

Since the domain of is , this formula is valid for .

# Find the derivative of arcsin (x/2)

Find the derivative of the function

Using the chain rule and the derivative of arcsine we have

This is valid for (since the domain of is ).

# Prove that arccot x – arctan (1/x) is not constant but has zero derivative

1. Prove that for we have

2. Prove that there is no constant such that for all we have

Why is this not a violation of the zero-derivative theorem (Theorem 5.2 in Apostol)?

1. Proof. We can use the formulas for the derivatives of and (and the chain rule) to compute,

2. Proof. First, let . Then,

Then, since , we have

Next, let . Then, Again, using that , we have

Hence, there is no constant such that for all

This is not a violation of the zero-derivative theorem since the function is constant on every open interval on which it is defined. Since it isn’t defined at , any open subinterval must be a subinterval of only positive or only negative reals. The function is constant on any of these subintervals.

# Establish the formula for the derivative of arccsc x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Where we use the trig identity in the final line. Then, since we have,

# Establish the formula for the derivative of arcsec x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Where we use the trig identity in the final line. Then, since we have,

# Establish the formula for the derivative of arccot x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Using a trig identity for tangent and cosecant we have

since . Therefore we conclude,

# Establish the formula for the derivative of arctan x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Using a trig identity for tangent and secant we have

since . Therefore we conclude,

# Establish the formula for the derivative of arccos x

Establish the following formula for the derivative of is correct,

For let

Then we know

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

Using the pythagorean identity for sine and cosine we have

since . Furthermore, since on the range of (i.e., for ) we must take the positive square root. Therefore we conclude,