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}} $
 

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