# Derivative formulas

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The derivative of any polynomial function of one variable is easily obtained. If $c\in\R$ (or a constant function) and $f,g:D\to\R$ are both differentiable on some set $D'$ , then so are $c\cdot f$ , $f+g$ , $-f$ , and $f\cdot g$ . If, in addition, $g$ is nonzero on $D'$ , then $\frac{1}{g}$ (and also $\frac{f}{g}$) are differentiable on $D'$ . Also, if $f$ is differentiable on $g\left(D'\right)$, then $f\circ g$ is differentiable on $D'$ . For the trivial case of $f(x)=a$ , for some constant $a$ (a degree 0 polynomial):

$a'=0$ [Proof]

For any $r\in\R$ :

$(x^r)'=rx^{r-1}$ [Proof]

Which covers any single variable polynomial function. Derivatives of non-polynomial functions require additional rules.

For any real-valued differentiable functions $f(x),g(x)$ :

• $(a\cdot f(x))'=a\cdot f'(x)$ [Proof]
• $(f(x)\pm g(x))'=f'(x)\pm g'(x)$ [Proof]
• $(-f(x))'=-f'(x)$
• $(f(x)\cdot g(x))' =f'(x)g(x)+f(x)g'(x)$ (Product rule)
• $\left(\frac{1}{g(x)}\right)'=-\frac{g'(x)}{g(x)^2}$
• $\left(\frac{f(x)}{g(x)}\right)'=\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^2}$ (Quotient rule)
• $(f\circ g)'=\bigl(f(g(x))\bigr)'=(f'(g))\cdot g'(x)$ (Chain rule)
• $\frac{d}{dx}(\sin(x))=\cos(x)$ [Proof]
• $\frac{d}{dx}(\cos(x))=-\sin(x)$
• $\frac{d}{dx}(\tan(x))=\sec^2(x)$
• $\frac{d}{dx}(\csc(x))=-\csc(x)\cot(x)$
• $\frac{d}{dx}(\sec(x))=\sec(x)\tan(x)$
• $\frac{d}{dx}(\cot(x))=-\csc^2(x)$
• $\frac{d}{dx}(\arcsin(x))=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\arccos(x))=-\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\arctan(x))=\frac{1}{1+x^2}$
• $\frac{d}{dx}(\arcsec(x))=\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}(\arccsc(x))=-\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}(\arccot(x))=-\frac{1}{1+x^2}$
• $\frac{d}{dx}(e^x)=e^x$
• $\frac{d}{dx}(a^x)=a^x\ln(a)$
• $\frac{d}{dx}(\ln(x))=\frac{1}{x}$ [Proof]
• $\frac{d}{dx}(\log_a(x))=\frac{1}{\ln(a)x}$