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The derivative of a given function represents a diminutive change in function with respect to one of its variables. We are familiar with seeing a derivative denoted such as:
\begin{equation*}
{{f}^{‘}(x)}
\end{equation*}
Or more formally
\begin{equation*}
{\frac{\normalsubformula{\text{df}}}{\normalsubformula{\text{dx}}}}
\end{equation*}
Which we can verbally state as “the derivative of the function ${f(x)}$ in respect to x”
Given one exists, a derivative of a given function can be taken n times, ${f^{{n}}(x)}$, such that:
\begin{equation*}
{\frac{d^{{n}}f}{\normalsubformula{\text{dx}}^{{n}}}}
\end{equation*}
The derivative of a function ${f(x)}$ can be iterated as such:
\begin{equation*}
{{f}^{‘}(x)=\underset{{h\rightarrow 0}}{{\text{lim}}}\frac{f(x+h)-f(x)}{h}}
\end{equation*}
If the first derivative exists, then the second derivative can be taken so that:
\begin{equation*}
{{f}^{”}(x)=\underset{{h\rightarrow 0}}{{\text{lim}}}\frac{{f}^{‘}(x+h)-{f}^{‘}(x)}{h}}
\end{equation*}
Note, that in order for a limit to exist, both the ${\underset{{h\rightarrow 0^{{+{}}}}}{{\text{lim}}}}$ and ${\underset{{h\rightarrow 0^{{-{}}}}}{{\text{lim}}}}$ must exist.
The following is a quick reference list of common derivatives that have been tried and tested throughout time and do not require that you go through the numerical differentiation to determine:
\begin{equation*}{\frac{d}{\normalsubformula{\text{dx}}}x^{{n}}=\normalsubformula{\text{nx}}^{{n-1}}}\end{equation*}
\begin{equation*}{\frac{d}{\normalsubformula{\text{dx}}}\text{ln}x=\frac{1}{x}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{sin}x=\text{cos}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{cos}x=-\text{sin}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{tan}x=\text{sec}^{{2}}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{csc}x=-\text{csc}x\text{cot}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{sec}x=\text{sec}x\text{tan}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{cot}x=-\text{csc}^{{2}}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}e^{{x}}=\normalsubformula{\text{xe}}^{{x}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}a^{{x}}=(\text{ln}a)a^{{x}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{sin}^{{-1}}x=\frac{1}{\sqrt{1-x^{{2}}}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{cos}^{{-1}}x=-{\frac{1}{\sqrt{1-x^{{2}}}}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{tan}^{{-1}}x=\frac{1}{1+x^{{2}}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{cot}^{{-1}}x=-{\frac{1}{1+x^{{2}}}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{sec}^{{-1}}x=\frac{1}{x\sqrt{1-x^{{2}}}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{csc}^{{-1}}x=-{\frac{1}{x\sqrt{1-x^{{2}}}}}}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{sinh}x=\text{cosh}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{cosh}x=\text{sinh}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{tanh}x=\text{sec}h^{{2}}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{coth}\normalsubformula{\text{hx}}=-\text{csc}h^{{2}}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{sec}\normalsubformula{\text{hx}}=-\text{sec}\normalsubformula{\text{hx}}\text{tanh}x}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\text{csc}\normalsubformula{\text{hx}}=-\text{csc}\normalsubformula{\text{hx}}\text{coth}x}
\end{equation*}
When dealing with a combination of functions, there are a number of important rules to adhere to when computing derivatives; they are as follows:
\begin{equation*}
{f(x)+\text{.}\text{.}\text{.}+z(x)={f}^{‘}(x)+\text{.}\text{.}\text{.}+{z}^{‘}(x)}
\end{equation*}
\begin{equation*}
{\frac{d}{\normalsubformula{\text{dx}}}\left[\normalsubformula{\text{cf}}(x)\right]=c{f}^{‘}(x)}
\end{equation*}
\begin{equation*}{\frac{d}{\normalsubformula{\text{dx}}}\left[f(x)g(x)\right]=f(x){g}^{‘}(x)+{f}^{‘}(x)g(x)}\quad [Product Rule]\end{equation*}
\begin{equation*}{\frac{d}{\normalsubformula{\text{dx}}}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x){f}^{‘}(x)-f(x){g}^{‘}(x)}{\left[g(x)\right]^{{2}}}} \quad [Quotient Rule]\end{equation*}
\begin{equation*}{\frac{d}{\normalsubformula{\text{dx}}}x^{{n}}=\normalsubformula{\text{nx}}^{{n-1}}} \quad [Power Rule]\end{equation*}
\begin{equation*}{\frac{d}{\normalsubformula{\text{dx}}}f(g(x))={f}^{‘}(g(x)){g}^{‘}(x)}\quad [Chain Rule] \end{equation*}
