# Fórmulas de Derivación

•   \cfrac{d}{dx} \ c = 0

• \cfrac{d}{dx} \ x = 1

• \cfrac{d}{dx} \ (u + v - w) = \cfrac{d}{dx}u + \cfrac{d}{dx}v \ - \cfrac{d}{dx}w

• \cfrac{d}{dx} \ (u \cdot v) = u' \cdot v+ v' \cdot u

#### La regla de la cadena

•   \cfrac{d}{dx} \ u^{n} = n \cdot u^{n-1} \cdot \cfrac{d}{dx}u

•  \cfrac{d}{dx} \ \sqrt{u} = \cfrac{\cfrac{d}{dx} u}{2 \cdot \sqrt{u}}

• \cfrac{d}{dx} \ \cfrac{u}{v} = \cfrac{u' \cdot v - v' \cdot u}{v^{2}}
• \cfrac{d}{dx} \ \cfrac{u}{c} = \cfrac{1}{c} \cdot \cfrac{d}{dx}u
• \cfrac{d}{dx} \ \cfrac{c}{u} = \cfrac{-c \cdot \cfrac{d}{dx}u}{u^{2}}

•   \cfrac{d}{dx} \ \sin u = \cos u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \cos u = -\sin u \cdot \cfrac{d}{dx}u

•  \cfrac{d}{dx} \ \tan u = \sec^{2}u \cdot \cfrac{d}{dx}u

•  \cfrac{d}{dx} \ \cot u = -\csc^{2}u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \sec u = \sec u \cdot \tan u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \csc u = - \csc u \cdot \cot u \cdot \cfrac{d}{dx}u

## Fórmulas de derivadas trigonométricas inversas

• \cfrac{d}{dx} \ \sin^{-1}u = \cfrac{\cfrac{d}{dx}u}{\sqrt{1-u^{2}}}   \qquad \bigg [ -\cfrac{\pi}{2} < \sin^{-1}u < \cfrac{\pi}{2} \bigg ]

• \cfrac{d}{dx} \ \cos^{-1}u = - \cfrac{\cfrac{d}{dx}u}{\sqrt{1-u^{2}}}   \qquad \bigg [ 0 < \cos^{-1} u < \pi \bigg ]

• \cfrac{d}{dx} \ \tan^{-1}u = \cfrac{\cfrac{d}{dx}u}{1+u^{2}} \qquad \bigg [ -\cfrac{\pi}{2} < \tan^{-1}u < \cfrac{\pi}{2} \bigg ]

• \cfrac{d}{dx} \ \cot^{-1}u = -\cfrac{\cfrac{d}{dx}u}{1+u^{2}} \qquad [0 < \cot^{-1} u < \pi ]

• \cfrac{d}{dx} \ \sec^{-1}u = \cfrac{\cfrac{d}{dx}u}{|u| \cdot \sqrt{u^{2} - 1}}

• \cfrac{d}{dx} \ \csc^{-1}u = - \cfrac{\cfrac{d}{dx}u}{|u| \cdot \sqrt{u^{2} - 1}}

## Fórmulas de derivadas exponenciales y logarítmicas

• \cfrac{d}{dx} \ \ln u = \cfrac{\cfrac{d}{dx}u}{u} = \cfrac{d}{dx} \log_{e} u
• \cfrac{d}{dx} \ e^{u} = e^{u} \cdot \cfrac{d}{dx}u
• \cfrac{d}{dx} \ \log_{a} u = \cfrac{\cfrac{d}{dx}u}{u \cdot \ln a} = \cfrac{\log_{a} e}{u} \cdot \cfrac{d}{dx}u
• \cfrac{d}{dx} \ u^{v} = \cfrac{d}{dx} e^{v \cdot \ln u} = e^{v \cdot \ln u } \cdot \cfrac{d}{dx}[v \cdot \ln u] = v \cdot u^{v-1} \cdot \cfrac{d}{dx}u + u^{v} \cdot \ln u \cfrac{d}{dx} v

• \cfrac{d}{dx} \ \sinh u = \cosh u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \cosh u = \sinh u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \tanh u = \text{sech}^{2} \ u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \coth u = -\text{csch}^{2} \ u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \text{sech} \ u = -\text{sech} \ u \cdot \tanh u \cdot \cfrac{d}{dx}u

• \cfrac{d}{dx} \ \text{csch} \ u = - \text{csch} \ u \cdot \text{coth} \ u \cdot \cfrac{d}{dx}u

## Fórmulas de derivadas hiperbólicas inversas

#### Derivada de seno hiperbólico inverso

• \cfrac{d}{dx} \ \sinh^{-1}u = \cfrac{1}{\sqrt{u^{2} + 1}} \cdot \cfrac{d}{dx}u

#### Derivada de coseno hiperbólico inverso

• \cfrac{d}{dx} \ \cosh^{-1}u = \cfrac{1}{\sqrt{u^{2} - 1}} \cdot \cfrac{d}{dx}u

#### Derivada de tangente hiperbólica inversa

• \cfrac{d}{dx} \ \tanh^{-1}u = \cfrac{1}{1 - u^{2}} \cdot \cfrac{d}{dx}u

#### Derivada de cotangente hiperbólica inversa

• \cfrac{d}{dx} \ \coth^{-1}u = \cfrac{1}{1 - u^{2}} \cdot \cfrac{d}{dx}u

#### Derivada de secante hiperbólica inversa

• \cfrac{d}{dx} \ \text{sech}^{-1}u = \cfrac{-1}{u \cdot \sqrt{1 - u^{2}}} \cdot \cfrac{d}{dx}u

#### Derivada de cosecante hiperbólica inversa

• \cfrac{d}{dx} \ \text{csch}^{-1}u = \cfrac{-1}{|u| \cdot \sqrt{1 + u^{2}}} \cdot \cfrac{d}{dx}u

## Representación de las derivadas de orden superior

\cfrac{d^{2}y}{dx^{2}} = f''(x) = y'' = \cfrac{d}{dx} \bigg ( \cfrac{dy}{dx} \bigg )

\cfrac{d^{3}y}{dx^{3}} = f'''(x) = y''' = \cfrac{d}{dx} \bigg ( \cfrac{d^{2}y}{dx^{2}} \bigg )