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EJERCICIOS DE DERIVADAS. GRANVILLE

1).y=x3
dydx=dx3dx=3(x)3-1=3x2
2).y=ax4-bx2
dydx=d(ax4-bx2)dx=d(ax4)dx-bdx2dx=4ax3-2bx
3).y=x43+5
dydx=dx43dx+d5dx=43x13 +0=43x13
4).y=3x35x2-7x3x4+87x3
y=3x3x25-7xx43+8x37=3x135-7x-13+8x37
dydx=3dx135dx-7dx-13dx+8d(x37)dx=3135(x85)-7-13x-43+8(37x-47)
=395x85+73x-43+247x-47
5).y=x2-35
dydx=dx2-35dx=5dx2-35-1dxdx2-3dx=5x2-34dx2dx+d3dx=5x2-34(2x)=10xx2-34
6). y=a2-x2
dydx=da2-x212dx=12a2-x212-1 da2-x2dx=12a2-x2-12 da2dx-dx2dx=12a2-x2-12 -2x
=-xa2-x2-12=-xa2-x212
=-xa2-x2

7).y=3x2+21+5x2
dydx=3x2+2d1+5x212dx+1+5x212d3x2+2dx
=3x2+2121+5x2-12 d1+5x2dx+1+5x2123d(x2)dx+d2dx
=3x2+2121+5x2-1210x+1+5x212 6x=3x2+25x1+5x2-12+6x1+5x212
=15x31+5x2-12+10x1+5x2-12+6x1+5x212=15x31+5x212+10x1+5x2126x1+5x2121=15x3+10x+6x1+5x221+5x2=15x3+10x+6x+30x31+5x2=45x3+16x1+5x2
8). y=a2+x2a2-x2=a2+x2a2-x212
dydx=a2-x212da2+x2dx-a2-x2da2-x212dx[a2-x212]2=a2-x212 2x-(a2+x2) [12a2-x2-12 da2-x2dx]a2-x2
=2xa2-x212-a2-x212a2-x2-12-2xa2-x2=2xa2-x212-a2-x2-xa2-x2-12a2-x2
=2xa2-x212-a2-x21-xa2-x212a2-x2=2xa2-x2121+xa2-x2a2-x212a2-x2=2xa2-x2+xa2+x2a2-x212a2-x2
=2a2x-2x3+a2x+x3a2-x212a2-x2= 3a2x-x3a2-x212a2-x21=3a2x-x3a2-x212a2-x2=3a2x-x3a2-x2329). y=3x4-2x2+8
dydx=d(3x4)dx-d2x2dx+d8dx=3d(x4)dx-2dx2dx+d8dx=34x2-22x+0= 12x3-4x
10). y=4+3x-2x3
dydx=d4dx+d3xdx-d2x3dx=0+3dxdx-2dx3dx=0+31-23x2 =3-6x2
11). ddtat5-5bt3

=ddtat5-ddt5bt3=addtt5-5bddtt3=5at4-15bt2

12). dydxz22-z77
=ddxz22-ddxz77=2z2-7z67=z-z6
13). dduu
=dduu12=12u21-1dduu=12u-121=12u-12=12u12=12uddu
14). ddx2x-3x2
=ddx2x-ddx3x2=2-x-1-1-6x-2-1=2-x-2-6x-3=-2x2+6x315). ddt2t43-3t23
ddt2t43-ddt3t23=2ddtt43-3ddtt23=83t13-63t-13=83t13-2t-13
16). dxt2x34+4x-14
ddx2x34+ddx4x-14=2ddxx34+4x-14=64x-14-x-54

17).ddx (x2/3-a2/3) =23x-1/3
=ddxx23-1-ddx(a23-1)=23x-1/3
18). ddxa+bx+cx2x2
=xddxa+bx+cx2ddxxx2=x0+bddxx+cddxx2-a+bx+cx2x2=xb+2cx-a+bx+cx2x2
=bx+2cx2-a-bx-cx2x2=cx2-ax2=cx2x2-ax2=c-ax2
19). y=x2-2x
y=x122-2x12=ddxx12-12-x12ddx2-2ddxx12x122=1x-1222x120-212x12x122=1x-1240-212x12-1x122ddx
=14x-12+x-12x=14x12+1xx12=14x+1xx
20). S=a+bt+ct2t12
dsdt=a+bt+ct2t-12)=a+bt+ct2ddtt-12+t-12ddta+bt+ct2
=a+bt+ct2-12t-32+t-12ddta+ddtbt+ddt(ct2)
=-at-322-bt-122-12ct12+bt-12+2ct12=-at-322-12bt-12+32ct12=-a2t32+b2t12+3t122=-a2t+ab2t+3ct2
21).y=ax+aax

12a(x)-1/2+[a(-12(ax)-3/2(ddxax))]=√ax . √ax
12a(x)-1/2+[a(-12(ax)-3/2a)12a(x)-1/2+[a(-12(ax)-3/2)
-12a2(ax)-3/2
a2√xx+-a22ax32=a2√x+[a22ax√ax
=a2√x+a2x√ax

22).r=1-2θ drdθ = - 11-2θ
r=1-2θ12
drdθ=121-2θ12-1ddθ(1-2θ)
drdθ=121-2ϴ-12ddϴ10-ddϴ2ϴ
drdϴ=121-2ϴ-12o-2ddθθ
drdθ=121-2θ-12o-21
drdθ=121-2θ-2 -21
drdθ=1-2θ-1/2-1
drdθ=11-2θ1/2
drdθ=-11-2θ

23).ft=(2-3t2)2

ft=2-3 t23
ddt2-3t23
-ddt3t2-23
-3 3t2-22ddt3t2-2
-33t2-22ddt-2+ddt3t2
-33t2-22ddt3t2-33t2-22.9t
ft= -18t2-3t22
26.

28). y=(a+ +bx2 )3
Y=3(a+bx2 )3-1 dydx (a+ bx2 )
x2 ddx (b)-b ddxx2
x2(0)-b (2X)
x4
0-2bx
x4
3(-2 bxx4 ) (a+ bx2 )2
-6 bxx4 (a+ bx2 )2
-6b (1) (a+ bx2 )2x4
x4-1
dydx =- 6bx2 (a+ bx2 ).

30).S=t∂2+τ2
S=t∂2+t212
dsdt=td∂2+t212dt+∂2+t2dtdx=t 12a2+t212 d ∂2+t2+∂2+t21
=t 12∂2+t2-122 t+α2+t2=t2α2+t2-12+α2+t2=2t2+α2a2+t231).y=a-xa+x
dydx a-xa+x(a+x) dydxa-x-a-xdydx(a+x)(a+x)2
a+x-1-a-x)(1)(a+x)2
-a-x-(a-x)-a-x-a+x=-2a(a+x)2

32).Y=x25-2x
Y= (x2)(5-2x)1/2
dydx=x2d5-2x12dx+5-2x12dx2dx=x2125-2x12d5-2xdx+5-2x12(2x)
=x2[12x-2x)12-2x+(5-2x)122x=x2[-xx-2x)12+(5-2x)12(2x)
=-x2+10x-4x25-2x=-5x2+10x5-2x

35).r=θ23-4θ
r=θ2 (3-4θ)12
drdθ =θ2 d(3-4θ)12dθ +3-4θd(θ2)dθ =θ2 12 (3-4θ)-12 (-4)+6θ-8θ2
=θ2-2(3-4θ)-1/2+6θ-8θ2=2θ2(3-4θ)1/2 +6θ-8θ2=6θ-10θ23-4θ
47).Y=a2+x2a2-x2
dydx=a2-x2da2+x2dx-(a2+x2)[da2-x2dx](a-x2)2
dydx=a2-x20+2x-(a2+x2)(0-2x)(a-x2)2
=a2-x22x-(a2+x2)(-2x)(a-x2)2
=2a2x-2x3+2a2x+2x3(a-x2)2
dydx=4a2x(a-x2)2
36). y=1-cx1+cx
y=1-cx1+cx12
dydx1-cx1+cx12=121-cx1+cx12-1ddx1-cx1+cx
y=121-cx1+cx-121+cxd1-cxdx-1-cxd1+cxdx1+cx2
y=121-cx1+cx-121+cxddx1-cddxx-1-cxddx1+cddxx1+cx2...