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Derivointikaavoja

(1) $$\begin{split}\begin{array}{cc|cc|cc}\hline f(x) & f'(x) & f(x) & f'(x) & f(x) & f'(x) \\\hline x^a & ax^{a - 1} & \sin x & \cos x & \sinh x & \cosh x \\[2ex] x^{\frac{1}{a}} & \dfrac{x^{\frac{1}{a} - 1}}{a} & \cos x & -\sin x & \cosh x & \sinh x \\[2ex] e^x & e^x & \tan x & \dfrac{1}{\cos^2 x} & \tanh x & \dfrac{1}{\cosh^2 x} \\[2ex] a^x & a^x\ln a & \arcsin x & \dfrac{1}{\sqrt{1 - x^2}} & \arsinh x & \dfrac{1}{\sqrt{1 + x^2}} \\[2ex] \ln x & \dfrac{1}{x} & \arccos x & -\dfrac{1}{\sqrt{1 - x^2}} & \arcosh x & \dfrac{1}{\sqrt{x^2 - 1}} \\[2ex] \log_a x & \dfrac{1}{x\ln a} & \arctan x & \dfrac{1}{1 + x^2} & \artanh x & \dfrac{1}{1 - x^2} \\[2ex]\hline \end{array}\end{split}$$

$$\begin{split}\begin{array}{cl}\hline \text{Kaava} & \text{Nimi} \\\hline D(cf(x)) = cf'(x) & \text{vakion siirto} \\ D(f(x) \pm g(x)) = f'(x) \pm g'(x) & \text{lineaarisuus} \\ D(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) & \text{tulon derivointi} \\[3ex] D\left(\dfrac{f(x)}{g(x)}\right) = \dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} & \text{osamäärän derivointi} \\[3ex] D((f \circ g)(x)) = f'(g(x))g'(x) & \text{ketjusääntö} \\[3ex] D(f^{-1}(y)) = \dfrac{1}{f'(x)}, \text{ kun } f(x) = y & \text{käänteisfunktion derivointi} \\[3ex]\hline \end{array}\end{split}$$

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