Poziom studiów
Najważniejsze wzory
\[ \begin{aligned} (c)' &= 0 \\[6pt] (x^n)' &= n\,x^{n-1} \\[6pt] (e^x)' &= e^x \\[6pt] (a^x)' &= a^x \ln a \\[6pt] (\ln x)' &= \frac{1}{x} \\[6pt] (\log_a x)' &= \frac{1}{x\,\ln a} \\[6pt] (\sin x)' &= \cos x \\[6pt] (\cos x)' &= -\sin x \\[6pt] (\operatorname{tg} x)' &= \frac{1}{\cos^2 x} \end{aligned} \]
\[ \begin{aligned} (\operatorname{ctg} x)' &= -\frac{1}{\sin^2 x} \\[6pt] (\operatorname{arctg} x)' &= \frac{1}{1 + x^2} \\[6pt] (\operatorname{arcctg} x)' &= -\frac{1}{1 + x^2}\\[6pt] (\arcsin x)' &= \frac{1}{\sqrt{1 - x^2}} \\[6pt] (\arccos x)' &= -\frac{1}{\sqrt{1 - x^2}} \\[6pt] (\sinh x)' &= \cosh x \\[6pt] (\cosh x)' &= \sinh x \\[6pt] \end{aligned} \]
Ze wzoru \((x^n)' = n\,x^{n-1} \) wynikają inne popularne wzory: \[ \begin{aligned} (\sqrt{x})'&=\frac{1}{2 \sqrt{x}} \\[6pt] \left(\frac{1}{x}\right)' &= -\frac{1}{x^2} \\[6pt] \end{aligned} \]
Najważniejsze reguły liczenia pochodnych
\[\bigl(c\cdot f(x)\bigl)'=c\cdot f'(x)\] \[\bigl(f(x)+g(x)\bigl)'=f'(x)+g'(x)\] \[\bigl(f(x)\cdot g(x)\bigl)'=f'(x)\cdot g(x)+f(x)\cdot g'(x)\] \[\left(\frac{f(x)}{g(x)}\right)'=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^2}\] \[ \Bigl(g\bigl(f(x)\bigl)\Bigl)^{\prime}=g'\bigl(f\left(x\right)\bigl) \cdot f'(x) \]