# Derivatives Formulas

## Basic forms and properties of derivatives

#### Derivative of a constant

• \cfrac{d}{dx} \ c = 0

#### Derivative of x

• \cfrac{d}{dx} \ x = 1

#### Derivative of a sum

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

#### Derivative of a multiplication

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

#### The chain rule

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

#### Derivative of a square root

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

#### Derivative of a division

• \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}}

## Trigonometric derivative formulas

#### Derivative of sine

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

#### Derivative of cosine

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

#### Derivative of tangent

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

#### Derivative of cotangent

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

#### Derivative of secant

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

#### Derivative of cosecant

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

## Inverse trigonometric derivative formulas

#### Derivative of sine inverse

• \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 ]

#### Derivative of cosine inverse

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

#### Derivative of tangent inverse

• \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 ]

#### Derivative of cotangent inverse

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

#### Derivative of secant inverse

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

#### Derivative of cosecant inverse

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

## Exponential and logarithmic derivative formulas

• \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

## Hyperbolic derivative formulas

#### Derivative of hyperbolic sine

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

#### Derivative of hyperbolic cosine

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

#### Derivative of hyperbolic tangent

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

#### Derivative of hyperbolic cotangent

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

#### Derivative of hyperbolic secant

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

#### Derivative of hyperbolic cosecant

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

## Formulas of inverse hyperbolic derivatives

#### Derivative of inverse hyperbolic sine

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

#### Derivative of inverse hyperbolic cosine

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

#### Derivative of inverse hyperbolic tangent

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

#### Derivative of inverse hyperbolic cotangent

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

#### Derivative of inverse hyperbolic secant

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

#### Derivative of inverse hyperbolic cosecant

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

## Representation of higher order derivatives

• Second derivative

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

• Third derivative

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

• N-th derivative

\cfrac{d^{n}y}{dx^{n}} = f^{n}(x) = y^{n} = \cfrac{d}{dx} \bigg ( \cfrac{d^{n - 1}y}{dx^{n - 1}} \bigg )