נגזרת / נגזרות של פונקציות אלנמטריות 📥 הורד PDF נגזרות של פונקציות בסיסיותעריכה d d x c = 0 {\displaystyle {\frac {d}{dx}}c=0} d d x x = 1 {\displaystyle {\frac {d}{dx}}x=1} d d x x c = c x c − 1 {\displaystyle {\frac {d}{dx}}x^{c}=cx^{c-1}} (כאשר הביטויים x c {\displaystyle x^{c}} ו- c x c − 1 {\displaystyle cx^{c-1}} מוגדרים) נגזרות של פונקציות מעריכיות ולוגריתמיותעריכה d d x e x = e x {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}} d d x a x = a x ⋅ ln ( a ) {\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\cdot \ln(a)} d d x ln ( x ) = 1 x x > 0 {\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}}\qquad x>0} d d x log a ( x ) = 1 x ⋅ ln ( a ) x > 0 {\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\cdot \ln(a)}}\qquad x>0} נגזרות של פונקציות טריגונומטריותעריכה d d x sin ( x ) = cos ( x ) {\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)} d d x cos ( x ) = − sin ( x ) {\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)} d d x tan ( x ) = sec 2 ( x ) = 1 cos 2 ( x ) {\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}} d d x sec ( x ) = tan ( x ) ⋅ sec ( x ) {\displaystyle {\frac {d}{dx}}\sec(x)=\tan(x)\cdot \sec(x)} d d x cot ( x ) = − csc 2 ( x ) = − 1 sin 2 ( x ) {\displaystyle {\frac {d}{dx}}\cot(x)=-\csc ^{2}(x)=-{\frac {1}{\sin ^{2}(x)}}} d d x csc ( x ) = − csc ( x ) ⋅ cot ( x ) {\displaystyle {\frac {d}{dx}}\csc(x)=-\csc(x)\cdot \cot(x)} d d x arcsin ( x ) = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}} d d x arccos ( x ) = − 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}} d d x arctan ( x ) = 1 x 2 + 1 {\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{x^{2}+1}}} d d x arcsec ( x ) = 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arcsec}(x)={\frac {1}{|x|{\sqrt {x^{2}-1}}}}} d d x arccot ( x ) = − 1 x 2 + 1 {\displaystyle {\frac {d}{dx}}\operatorname {arccot}(x)=-{\frac {1}{x^{2}+1}}} d d x arccsc ( x ) = − 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arccsc}(x)=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}} נגזרות של פונקציות היפרבוליותעריכה d d x sinh ( x ) = cosh ( x ) = e x + e − x 2 {\displaystyle {\frac {d}{dx}}\sinh(x)=\cosh(x)={\frac {e^{x}+e^{-x}}{2}}} d d x cosh ( x ) = sinh ( x ) = e x − e − x 2 {\displaystyle {\frac {d}{dx}}\cosh(x)=\sinh(x)={\frac {e^{x}-e^{-x}}{2}}} d d x tanh ( x ) = sech 2 ( x ) {\displaystyle {\frac {d}{dx}}\tanh(x)=\operatorname {sech} ^{2}(x)} d d x sech ( x ) = − tanh ( x ) ⋅ sech ( x ) {\displaystyle {\frac {d}{dx}}\operatorname {sech} (x)=-\tanh(x)\cdot \operatorname {sech} (x)} d d x coth ( x ) = − csch 2 ( x ) {\displaystyle {\frac {d}{dx}}\operatorname {coth} (x)=-\operatorname {csch} ^{2}(x)} d d x csch x = − coth ( x ) ⋅ csch ( x ) {\displaystyle {\frac {d}{dx}}\operatorname {csch} x=-\operatorname {coth} (x)\cdot \operatorname {csch} (x)} d d x arsinh ( x ) = 1 x 2 + 1 {\displaystyle {\frac {d}{dx}}\operatorname {arsinh} (x)={\frac {1}{\sqrt {x^{2}+1}}}} d d x arcosh ( x ) = 1 x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arcosh} (x)={\frac {1}{\sqrt {x^{2}-1}}}} d d x artanh ( x ) = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\operatorname {artanh} (x)={\frac {1}{1-x^{2}}}} d d x arsech ( x ) = − 1 x 1 − x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arsech} (x)=-{\frac {1}{x{\sqrt {1-x^{2}}}}}} d d x arcoth ( x ) = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arcoth} (x)={\frac {1}{1-x^{2}}}} d d x arcsch ( x ) = − 1 | x | x 2 + 1 {\displaystyle {\frac {d}{dx}}\operatorname {arcsch} (x)=-{\frac {1}{|x|{\sqrt {x^{2}+1}}}}} מקור: ויקיספר העברי · רישיון CC BY-SA 4.0 · התוכן עובד והותאם → ממשים ושדה שלם נגזרת / נגזרת של פונקציה הפיכה ←
