導関数チートシート

導関数の規則

冪乗則 ddx (x^a)=a·x^a−1

定数の導関数 ddx (a)=0

和差法則 (f±g)^′=f^′±g^′

定数除外 (a·f)^′=a·f^′

積の法則 (f·g)^′=f^′·g+f·g^′

商の法則 (fg )^′=f^′·g−g^′·fg²

連鎖律 df(u)dx =dfdu ·dudx

ddx (ln(x))=1x ddx (ln(|x|))=1x

ddx (e^x)=e^x ddx (log(x))=1xln(10)

ddx (log_a(x))=1xln(a)

ddx (sin(x))=cos(x) ddx (cos(x))=−sin(x)

ddx (tan(x))=sec²(x) ddx (sec(x))=tan(x)cos(x)

ddx (csc(x))=−cot(x)sin(x) ddx (cot(x))=−1sin²(x)

ddx (arcsin(x))=1√1−x² ddx (arccos(x))=−1√1−x²

ddx (arctan(x))=1x²+1 ddx (arcsec(x))=1√x²√x²−1

ddx (arccsc(x))=−1√x²√x²−1 ddx (arccot(x))=−1x²+1

ddx (sinh(x))=cosh(x) ddx (cosh(x))=sinh(x)

ddx (tanh(x))=sech²(x) ddx (sech(x))=tanh(x)(−sech(x))

ddx (csch(x))=−coth(x)csch(x) ddx (coth(x))=−csch²(x)

ddx (arcsinh(x))=1√x²+1 ddx (arccosh(x))=1√x−1√x+1

ddx (arctanh(x))=11−x² ddx (arcsech(x))=√2x+1 −1(x−1)x

ddx (arccsch(x))=−1√1x² +1x² ddx (arccoth(x))=11−x²