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Differential Calculus Tables

Remember that \( f^{\prime} = \frac{df}{dx} \).

Rules

\(\left( f + g \right)^{\prime} = f^{\prime} + g^{\prime}\)
\(\left(f - g\right)^{\prime} = f^{\prime} - g^{\prime}\)
\(\left(c f\right)^{\prime} = c f^{\prime}\)
\(\left(f g\right)^{\prime} = f^{\prime} g + f g^{\prime}\)
\(\left( \frac{f}{g} \right)^{\prime} = \frac{f^{\prime} g - f g^{\prime}}{g^{2}},\ g \neq 0\)
\(\left( \frac{1}{f} \right)^{\prime} = \frac{- f^{\prime}}{f^{2}},\ f \neq 0\)
\(f\left( g \left( x \right)\right)^{\prime} = \left( f \circ g \right)^{\prime} = \frac{df}{dg} \frac{dg}{dx} = f^{\prime}\left( g \right) g^{\prime}\)
\(\left(f\left( x \right) g \left( x \right)\right)^{\left(n\right)} = \sum_{i = 0}^{n}{n \choose i} f^{\left( n - i \right)}\left(x \right) g^{\left(i \right)}\left(x \right)\)

\(c^{\prime} = 0\)
\(x^{\prime} = 1\)
\(\left(cx\right)^{\prime} = c\)
\(\left(x^{n}\right)^{\prime} = nx^{n - 1}\)
\(\left(\left( cx \right)^{n}\right)^{\prime} = ncx^{n - 1}\)
\(\vert x \vert^{\prime} = \frac{x}{\vert x \vert} = \operatorname{sgn} x,\ x \neq 0\)
\(\left(\frac{1}{x}\right)^{\prime} = \left(x^{-1}\right)^{\prime} = -x^{-2} = -\frac{1}{x^{2}}\)
\(\left(\frac{1}{x^{c}}\right)^{\prime} = \left(x^{-c}\right)^{\prime} = -cx^{-c-1} = -\frac{c}{x^{c+1}}\)
\(\left(\sqrt{x}\right)^{\prime} = \left(x^{\frac{1}{2}}\right)^{\prime} = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{ 2 \sqrt{x}},\ x>0\)
\(\left(\sqrt[m]{x^{n}}\right)^{\prime} = \left(x^{\frac{n}{m}}\right)^{\prime} = \frac{n}{m} x^{-\left(\frac{n}{m} - 1\right)} = \frac{n}{ m \sqrt[m]{x^{-n+m}}},\ x>0\)
\(\left( f^{-1} \right)^{\prime} = \frac{1}{f^{\prime} \circ f^{-1}}\)

\(\left(e^{x}\right)^{\prime} = e^{x}\)
\(\left(e^{cx}\right)^{\prime} = ce^{cx}\)
\(\left(c^{x}\right)^{\prime} = c^{x}\ln(c); \; c > 0\)
\(\left(\log_{c}x\right)^{\prime} = \frac{1}{x \ln(c)}\; ; \; c > 0, \; c\neq 1\)
\(\left(\ln x\right)^{\prime} = \log_{e}x = \frac{1}{x}\)
\(\left(\ln \vert x \vert\right)^{\prime} = \frac{1}{x}\)
\(\left(x^{x}\right)^{\prime} = x^{x} \left( 1 + \ln x \right)\)
\(\left(f^{g}\right)^{\prime} = e^{g \ln(f)} = f^{g} \left( f^{\prime} \frac{g}{f} + g^{\prime} \ln(f) \right) , \; f > 0\)
\(\left(c^{f}\right)^{\prime} = e^{f \ln(c)} = c^{f} f^{\prime} \ln (c)\)
\(\left(f^{g}\right)^{\prime} = f^{g} \left( f^{\prime} \frac{g}{f} + g^{\prime} \ln f \right),\ f\left( x \right) > 0\)

\(\left(\arcsin(x)\right)^{\prime} = \frac{1}{\sqrt{1 - x^{2}}}\)
\(\left(\arccos(x)\right)^{\prime} = -\frac{1}{\sqrt{1 - x^{2}}}\)
\(\left(\arctan(x)\right)^{\prime} = \frac{1}{x^{2} + 1}\)
\(\left(\operatorname{arccot}(x)\right)^{\prime} = - \frac{1}{x^{2} + 1}\)
\(\left(\operatorname{arcsec}(x)\right)^{\prime} = \frac{1}{ x \sqrt{x^{2} - 1}}\)
\(\left(\operatorname{arccsc}(x)\right)^{\prime}s = - \frac{1}{ x \sqrt{x^{2} - 1}}\)

\(\left(\sinh(x)\right)^{\prime} = \cosh(x)\)
\(\left(\cosh(x)\right)^{\prime} = \sinh(x)\)
\(\left(\tanh(x)\right)^{\prime} = \operatorname{sech}^{2}(x)\)
\(\left(\operatorname{sech}(x)\right)^{\prime} = -\tanh(x)\operatorname{sech}(x)\)
\(\left(\coth(x)\right)^{\prime} = -\operatorname{csch}^{2}(x)\)
\(\left(\operatorname{csch}(x)\right)^{\prime} = -\coth(x)\operatorname{csch}(x)\)

\(\left(\operatorname{arsinh}(x)\right)^{\prime} = \frac{1}{\sqrt{x^{2} + 1}}\)
\(\left(\operatorname{arcosh}(x)\right)^{\prime} = \frac{1}{\sqrt{x^{2} - 1}} \; , \; x > 1\)
\(\left(\operatorname{artanh}(x)\right)^{\prime} = \frac{1}{1 - x^{2}} \; , \; \vert x \vert < 1\)
\(\left(\operatorname{arcsch}(x)\right)^{\prime} = -\frac{1}{\vert x \vert \sqrt{1 + x^{2}}} \; , \; x \neq 0\)
\(\left(\operatorname{arsech}(x)\right)^{\prime} = -\frac{1}{\vert x \vert \sqrt{1 - x^{2}}} \; , \; 0 < x < 1\)
\(\left(\operatorname{arcoth}(x)\right)^{\prime} = \frac{1}{1 - x^{2}} \; , \; \vert x \vert > 1\)