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41.
The reciprocal rule
$\left(\dfrac{1}{f}\right)' = -\dfrac{f'}{f^2}$
Differentiation
Differentiation rule
42.
The quotient rule
$\left(\dfrac{f}{g}\right)'=\dfrac{f'g-g'f}{f^2}$
Differentiation
Differentiation rule
43.
Generalized power rule
$\left(f^g\right)'=f^g\,\left( f'\,\dfrac{g}{f} + g' \ln f \right)$
Differentiation
Differentiation rule
44.
Logarithmic derivatives
$\left( \ln f \right)' = \dfrac{f'}{f}$
Differentiation
Differentiation rule
45.
The chain rule - general formula for composite functions
$\left(f\left(g(x)\right)\right)'=f'(g(x))\cdot g'(x)$
Differentiation
Differentiation rule
46.
$\left(f^2\right)'=2\cdot f \cdot f'$
Differentiation
47.
$\left( \sin f\right)'=\cos f \cdot f'$
Differentiation
48.
$\left( \cos f \right)' = - \sin f \cdot f'$
Differentiation
49.
$\left( \tan f \right)' = \dfrac{f'}{\cos f^2}$
Differentiation
50.
$\left( e^f \right)' = e^f \cdot f'$
Differentiation
51.
$\left( a^x\right)' = a^f \cdot \ln a \cdot f'$
Differentiation
52.
Derivative of inverse coth
$\dfrac{d}{dx}\, \left( \coth^{-1} x\right) = \dfrac{1}{1-x^2}$
Differentiation
53.
Linearity of differentiation
$\left(af+bg\right)'=af'+bg'$
Differentiation
Differentiation rule
54.
$\dfrac{d}{dx}\, \left(ax^2+bx+c\right) = 2ax+b$
Differentiation
55.
$\dfrac{d}{dx}\, \left( x^n \right) = nx^{n-1}$
Differentiation
56.
$\left(f^n\right)'=n \cdot f^{n-1} \cdot f'$
Differentiation
57.
$\left(\sqrt{f}\right)' = \dfrac{f'}{2\sqrt{f}}$
Differentiation
58.
$\left( \ln f \right)' = \dfrac{f'}{f}$
Differentiation
59.
$\left( \log_a x \right)'=\dfrac{f'}{f \ln a}$
Differentiation
60.
n-th derivative of $e^x$
$\left(e^x\right)^{(n)}=e^x$
Differentiation

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