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 )
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Derivatives form
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