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

## Maxima and Minima Using Trigonometric and Inverse Trigonometric Functions

Many problems in application of maxima and minima may be solved easily by making use of trigonometric or inverse trigonometric functions. The basic idea is the same; identify the constant terms and identify the variable to be maximized or minimized, differentiate that variable then equate to zero.