1)Tính đạo hàm cấp 10 của hàm số $y=(2x+3)\cos x^{2}$
2)Tính đạo hàm cấp 4 của hàm số $y=e^{-x^2}$
Giải giùm mình với thanks!!!
Nhìn mà nản !
1) $y=(2x+3)\cos (x^{2})$
$y'=2\cos \left( {x}^{2} \right) -2 \left( 2x+3 \right) \sin \left(
{x}^{2} \right) x
$
$y''=-8\sin \left( {x}^{2} \right) x-4 \left( 2x+3 \right) \cos
\left( {x}^{2} \right) {x}^{2}-2 \left( 2x+3 \right) \sin \left(
{x}^{2} \right) $
$y'''=-48\cos \left( {x}^{2} \right) {x}^{2}-12\sin \left( {x}^{2}
\right) +16\sin \left( {x}^{2} \right) {x}^{4}+24\sin \left( {x}^
{2} \right) {x}^{3}-36\cos \left( {x}^{2} \right) x
$
$y^{(4)}=160\sin \left( {x}^{2} \right) {x}^{3}-120\cos \left( {x}^{2}
\right) x+32\cos \left( {x}^{2} \right) {x}^{5}+48\cos \left( {x}
^{2} \right) {x}^{4}+144\sin \left( {x}^{2} \right) {x}^{2}-36\cos
\left( {x}^{2} \right)
$
$y^{(5)}=480\cos \left( {x}^{2} \right) {x}^{4}+720\sin \left( {x}^{2}
\right) {x}^{2}-120\cos \left( {x}^{2} \right) -64\sin \left( {x}
^{2} \right) {x}^{6}-96\sin \left( {x}^{2} \right) {x}^{5}+480\cos
\left( {x}^{2} \right) {x}^{3}+360\sin \left( {x}^{2} \right) x$
$y^{(6)}=-1344\sin \left( {x}^{2} \right) {x}^{5}+3360\cos \left( {x}^{2}
\right) {x}^{3}+1680\sin \left( {x}^{2} \right) x-128\cos \left(
{x}^{2} \right) {x}^{7}-192\cos \left( {x}^{2} \right) {x}^{6}-1440
\sin \left( {x}^{2} \right) {x}^{4}+2160\cos \left( {x}^{2}
\right) {x}^{2}+360\sin \left( {x}^{2} \right) $
$y^{(7)}=-3584\cos \left( {x}^{2} \right) {x}^{6}-13440\sin \left( {x}^{2}
\right) {x}^{4}+13440\cos \left( {x}^{2} \right) {x}^{2}+1680\sin
\left( {x}^{2} \right) +256\sin \left( {x}^{2} \right) {x}^{8}+384
\sin \left( {x}^{2} \right) {x}^{7}-4032\cos \left( {x}^{2}
\right) {x}^{5}-10080\sin \left( {x}^{2} \right) {x}^{3}+5040\cos
\left( {x}^{2} \right) x$
$y^{(8)}=9216\sin \left( {x}^{2} \right) {x}^{7}-48384\cos \left( {x}^{2}
\right) {x}^{5}-80640\sin \left( {x}^{2} \right) {x}^{3}+30240
\cos \left( {x}^{2} \right) x+512\cos \left( {x}^{2} \right) {x}^{9}
+768\cos \left( {x}^{2} \right) {x}^{8}+10752\sin \left( {x}^{2}
\right) {x}^{6}-40320\cos \left( {x}^{2} \right) {x}^{4}-40320
\sin \left( {x}^{2} \right) {x}^{2}+5040\cos \left( {x}^{2} \right) $
$y^{(9)}=23040\cos \left( {x}^{2} \right) {x}^{8}+161280\sin \left( {x}^{2}
\right) {x}^{6}-403200\cos \left( {x}^{2} \right) {x}^{4}-302400
\sin \left( {x}^{2} \right) {x}^{2}+30240\cos \left( {x}^{2}
\right) -1024\sin \left( {x}^{2} \right) {x}^{10}-1536\sin
\left( {x}^{2} \right) {x}^{9}+27648\cos \left( {x}^{2} \right) {x}
^{7}+145152\sin \left( {x}^{2} \right) {x}^{5}-241920\cos \left( {
x}^{2} \right) {x}^{3}-90720\sin \left( {x}^{2} \right) x$
$y^{(10)}=-56320\sin \left( {x}^{2} \right) {x}^{9}+506880\cos \left( {x}^{2
} \right) {x}^{7}+1774080\sin \left( {x}^{2} \right) {x}^{5}-2217600
\cos \left( {x}^{2} \right) {x}^{3}-665280\sin \left( {x}^{2}
\right) x-2048\cos \left( {x}^{2} \right) {x}^{11}-3072\cos
\left( {x}^{2} \right) {x}^{10}-69120\sin \left( {x}^{2} \right) {x
}^{8}+483840\cos \left( {x}^{2} \right) {x}^{6}+1209600\sin
\left( {x}^{2} \right) {x}^{4}-907200\cos \left( {x}^{2} \right) {x
}^{2}-90720\sin \left( {x}^{2} \right) $
(Áp dụng wolframalpha)
2. Tương tự !
$y={{\rm e}^{-{x}^{2}}}$
$y'=-2x{{\rm e}^{-{x}^{2}}}$
$y''=-2{{\rm e}^{-{x}^{2}}}+4{x}^{2}{{\rm e}^{-{x}^{2}}}$
$y'''=12x{{\rm e}^{-{x}^{2}}}-8{x}^{3}{{\rm e}^{-{x}^{2}}}$
$y^{(4)}=12{{\rm e}^{-{x}^{2}}}-48{x}^{2}{{\rm e}^{-{x}^{2}}}+16{x}^{4}{
{\rm e}^{-{x}^{2}}}=4{{\rm e}^{-{x}^{2}}} \left( 3-12{x}^{2}+4{x}^{4} \right) $