Derivative Formula

Derivatives are a fundamental tool of calculus. The derivative of a function of a real variable measures the sensitivity to change of a quantity, which is determined by another quantity.

Derivative Formula is given as,

f^{1} (x) = lim_{△ x \to 0} \frac{f (x + △ x) - f (x)}{△ x}

Some Basic Derivatives

\begin{array}{l} \frac{d}{d x} (c) = 0 \end{array}

\begin{array}{l} \frac{d}{d x} (x) = 1 \end{array}

\begin{array}{l} \frac{d}{d x} (x^{n}) = n x^{n - 1} \end{array}

\begin{array}{l} \frac{d}{d x} (u \pm v) = \frac{d u}{d x} \pm \frac{d v}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (c u) = c \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}} \end{array}

Chain Rule

\begin{array}{l} \frac{d}{d x} (u . v) = \frac{d v}{d x} (\frac{d u}{d x} . v) \end{array}

\begin{array}{l} \frac{d u}{d x} = \frac{d u}{d x} \frac{d v}{d x} \end{array}

Derivative of the Inverse Function

x(y) is the inverse of the function y(x),

\begin{array}{l} \frac{d y}{d x} = \frac{1}{\frac{d x}{d y}} \end{array}

Derivative of Trigonometric Functions and their Inverses

\begin{array}{l} \frac{d}{d x} (\sin (u)) = \cos (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\cos (u)) = - \sin (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\tan (u)) = \sec^{2} (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\cot (u)) = - \csc^{2} (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\sec (u)) = \sec (u) \tan (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\csc (u)) = - \csc (u) \cot (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\sin^{- 1} (u)) = \frac{1}{\sqrt{1 - u^{2}}} \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\cos^{- 1} (u)) = - \frac{1}{\sqrt{1 - u^{2}}} \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\tan^{- 1} (u)) = \frac{1}{1 + u^{2}} \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\cot^{- 1} (u)) = - \frac{1}{1 + u^{2}} \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\sec^{- 1} (u)) = \frac{1}{| u | \sqrt{u^{2} - 1}} \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\csc^{- 1} (u)) = - \frac{1}{| u | \sqrt{u^{2} - 1}} \frac{d u}{d x} \end{array}

Derivative of the Hyperbolic functions and their Inverses

\begin{array}{l} \frac{d}{d x} (\sinh (u)) = \cosh (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (\cosh (u)) = \sinh (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (t a n h (u)) = s e c h^{2} (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (c o t h (u)) = - c s c h^{2} (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (s e c h (u)) = - s e c h (u) t a n h (u) \frac{d u}{d x} \end{array}

\begin{array}{l} \frac{d}{d x} (c s c h (u)) = - c s c h (u) c o t h (u) \frac{d u}{d x} \end{array}

More topics in Derivative Formula
Antiderivative Formula	Chain Rule Formula
Instantaneous Rate of Change Formula	Product Rule Formula
Implicit Differentiation Formula	Mean Value Theorem Formula
Quotient Rule Formula	Newton’s Method Formula