# 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,
$\LARGE f^{1}(x)=\lim_{\triangle x \rightarrow 0}\frac{f(x+ \triangle x)-f(x)}{\triangle x}$

#### Some Basic Derivatives

$$\begin{array}{l}\large \frac{d}{dx}(c)=0\end{array}$$

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

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

$$\begin{array}{l}\large \frac{d}{dx}(u\pm v)=\frac{du}{dx}\pm \frac{dv}{dx}\end{array}$$

$$\begin{array}{l}\large \frac{d}{dx}(cu)=c\frac{du}{dx}\end{array}$$

$$\begin{array}{l}\large \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\end{array}$$

$$\begin{array}{l}\large \frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}\end{array}$$

#### Chain Rule

$$\begin{array}{l}\large \frac{d}{dx}(u.v)=\frac{dv}{dx}\left ( \frac{du}{dx}.v\right )\end{array}$$
$$\begin{array}{l}\large \frac{du}{dx}=\frac{du}{dx}\frac{dv}{dx}\end{array}$$

#### Derivative of the Inverse Function

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

$$\begin{array}{l}\large \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}\end{array}$$

#### Derivative of Trigonometric Functions and their Inverses

$$\begin{array}{l}\large \frac{d}{dx}(\sin (u))=\cos (u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\cos (u))=-\sin (u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\tan (u))=\sec^{2}(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\cot(u))=-\csc^{2}(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\sec(u))=\sec(u)\tan(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\csc(u))=-\csc(u)\cot(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\sin^{-1}(u))=\frac{1}{\sqrt{1-u^{2}}}\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\cos ^{-1}(u))=-\frac{1}{\sqrt{1-u^{2}}}\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\tan^{-1}(u))=\frac{1}{1+u^{2}}\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\cot^{-1}(u))=-\frac{1}{1+u^{2}}\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\sec ^{-1}(u))=\frac{1}{\left | u \right |\sqrt{u^{2}-1}}\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\csc^{-1}(u))=-\frac{1}{\left | u \right |\sqrt{u^{2}-1}}\frac{du}{dx}\end{array}$$

#### Derivative of the Hyperbolic functions and their Inverses

$$\begin{array}{l}\large \frac{d}{dx}(\sinh(u))=\cosh(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(\cosh(u))=\sinh(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(tanh(u))=sech^{2}(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(coth(u))=-csch^{2}(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(sech(u))=-sech(u)tanh(u)\frac{du}{dx}\end{array}$$
$$\begin{array}{l}\large \frac{d}{dx}(csch(u))=-csch(u)coth(u)\frac{du}{dx}\end{array}$$

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