# Differential Calculus

In the following formulas,   $u$,   $v$,   and $w$   are differentiable functions of   $x$   and   $a$   and   $n$   are constants.

Differentiation of Algebraic Functions

1. $\dfrac{d}{dx}(c) = 0$

2. $\dfrac{d}{dx}(x) = 1$

3. $\dfrac{d}{dx}(u) = \dfrac{du}{dx}$

4. $\dfrac{d}{dx}(cu) = c ~ \dfrac{du}{dx}$

5. $\dfrac{d}{dx}(u + v) = \dfrac{du}{dx} + \dfrac{dv}{dx}$

6. $\dfrac{d}{dx}(uv) = u ~ \dfrac{dv}{dx} + v ~ \dfrac{du}{dx}$

7. $\dfrac{d}{dx}(u^n) = nu^{n - 1} ~ \dfrac{du}{dx}$

8. $\dfrac{d}{dx}(\sqrt{u}) = \dfrac{\dfrac{du}{dx}}{2\sqrt{u}}$

9. $\dfrac{d}{dx}\left( \dfrac{u}{v} \right) = \dfrac{v ~ \dfrac{du}{dx} - u ~ \dfrac{dv}{dx}}{v^2}$

10. $\dfrac{d}{dx}\left( \dfrac{c}{v} \right) = \dfrac{-c ~ \dfrac{dv}{dx}}{v^2}$

11. $\dfrac{dy}{dx} = \dfrac{1}{\dfrac{dx}{dy}} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} = \dfrac{\dfrac{dy}{du}}{\dfrac{dx}{du}}$

Differentiation of Logarithmic and Exponential Functions

1. $\dfrac{d}{dx}(\log_a u) = \dfrac{\dfrac{du}{dx}}{u \ln a}$

2. $\dfrac{d}{dx}(\log u) = \dfrac{\log e ~ \dfrac{du}{dx}}{u} = \dfrac{\dfrac{du}{dx}}{u \ln 10}$

3. $\dfrac{d}{dx}(\ln u) = \dfrac{\dfrac{du}{dx}}{u}$

4. $\dfrac{d}{dx}(a^u) = a^u \ln a ~ \dfrac{du}{dx}$

5. $\dfrac{d}{dx}(e^u) = e^u \dfrac{du}{dx}$

6. $\dfrac{d}{dx}(u^v) = vu^{v - 1} ~ \dfrac{du}{dx} + u^v \, \ln u ~ \dfrac{dv}{dx}$

Differentiation of Trigonometric Functions

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

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. $\dfrac{d}{dx}({\rm arccsc} ~ u) = \dfrac{- ~ \dfrac{du}{dx}}{u\sqrt{u^2 - 1}}$

Differentiation of Hyperbolic Functions

1. $\dfrac{d}{dx}(\sinh \, u) = \cosh \, u ~ \dfrac{du}{dx}$

2. $\dfrac{d}{dx}(\cosh \, u) = \sinh \, u ~ \dfrac{du}{dx}$

3. $\dfrac{d}{dx}(\tanh \, u) = \text{sech}^2 \, u ~ \dfrac{du}{dx}$

4. $\dfrac{d}{dx}(\coth \, u) = -\text{csch}^2 \, u ~ \dfrac{du}{dx}$

5. $\dfrac{d}{dx}({\rm sech} \, u) = -\text{sech} \, u \tanh u ~ \dfrac{du}{dx}$

6. $\dfrac{d}{dx}({\rm csch} \, u) = -\text{csch} \, u \coth u ~ \dfrac{du}{dx}$

Differentiation of Inverse Hyperbolic Functions

1. $\dfrac{d}{dx}({\rm arcsinh} ~ u) = \dfrac{\dfrac{du}{dx}}{\sqrt{u^2 + 1}}$

2. $\dfrac{d}{dx}({\rm arccosh} ~ u) = \dfrac{\dfrac{du}{dx}}{\sqrt{u^2 - 1}}$

3. $\dfrac{d}{dx}({\rm arctanh} ~ u) = \dfrac{\dfrac{du}{dx}}{1 - u^2}$

4. $\dfrac{d}{dx}({\rm arccoth} ~ u) = \dfrac{\dfrac{du}{dx}}{1 - u^2}$

5. $\dfrac{d}{dx}({\rm arcsech} ~ u) = \dfrac{- ~ \dfrac{du}{dx}}{u\sqrt{1 - u^2}}$

5. $\dfrac{d}{dx}({\rm arccsch} ~ u) = \dfrac{- ~ \dfrac{du}{dx}}{u\sqrt{1 + u^2}}$

Chapter 1 - Fundamentals
Chapter 2 - Algebraic Functions
Chapter 3 - Applications
Chapter 4 - Trigonometric and Inverse Trigonometric Functions
Chapter 5 - Logarithmic and Exponential Functions
Miscellaneous