# Chapter 4 - Trigonometric and Inverse Trigonometric Functions

**Differentiation of Trigonometric Functions**

Trigonometric identities and formulas are basic requirements for this section. If *u* is a function of *x*, then

1. $\dfrac{d}{dx}(\sin \, u) = \cos \, u \dfrac{du}{dx}$

2. $\dfrac{d}{dx}(\cos \, u) = -\sin \, u \dfrac{du}{dx}$

3. $\dfrac{d}{dx}(\tan \, u) = \sec^2 \, u \dfrac{du}{dx}$

4. $\dfrac{d}{dx}(\cot \, u) = -\csc^2 \, u \dfrac{du}{dx}$

5. $\dfrac{d}{dx}(\sec \, u) = \sec \, u \, \tan \, u \dfrac{du}{dx}$

6. $\dfrac{d}{dx}(\csc \, u) = -\csc \, u \, \cot \, u \dfrac{du}{dx}$

**Differentiation of Inverse Trigonometric Functions**

In the formula below, *u* is any function of *x*.

1. $\dfrac{d}{dx}\arcsin \, u = \dfrac{\dfrac{du}{dx}}{\sqrt{1 - u^2}}$

2. $\dfrac{d}{dx}\arccos \, u = -\dfrac{\dfrac{du}{dx}}{\sqrt{1 - u^2}}$

3. $\dfrac{d}{dx}\arctan \, u = \dfrac{\dfrac{du}{dx}}{1 + u^2}$

4. $\dfrac{d}{dx}{\rm arccot} \, u = -\dfrac{\dfrac{du}{dx}}{1 + u^2}$

5. $\dfrac{d}{dx}{\rm arcsec} \, u = \dfrac{\dfrac{du}{dx}}{u\sqrt{u^2 - 1}}$

6. $\dfrac{d}{dx}{\rm arccsc} \, u = -\dfrac{\dfrac{du}{dx}}{u\sqrt{u^2 - 1}}$