Derivadas

a

1) G(r)=52r2-2

r-1 dG(r)

dr=dG(r)

dudu

dru = 2r2-2

r-1du

dr=(2r2-2)'(r-1)-(2r2-2)(r-1)'

(r-1)2du

dr=4r(r-1)-(2r2-2)

(r-1)2=4r2-4r-2r2+2

(r-1)2=2r2-4r+2

(r-1)2=2(r2-2r+1)

(r2-2r+1)=2dG

du=d

du(u1/5)=1

5u1

5-5

5=1

5u-4/5=1

5u4/5=1

55u4dG(r)

du=2

51

5(2r2-2

r-1)4, com r≠1ou ainda considerando os produtos notáveis, de tal modo que:2r2-2=2(r2-1)=2(r+1)(r-1)u(r)=2r2-2

r-1=2(r+1)(r-1)

(r-1)=2(r+1)du

dr=2G(u)=5udG

du=1

5u4/5=1

55u4dG

dr=dG

dudu

dr→dG

dr=1

55u4·2=2

5·1

5[2(r+1)]4 2) M(x)=x+x+xM(x)=(x+(x+x1/2)1/2)1/2dM

dx=1

2(x+(x+x1/2)1/2)1/2d

dx(x+x+x)dM

dx=1

2x+x+x(1 +1

2x+x+1

4xx+x)dM

dx=1

x+x+x(1

2+1

4x+x+1

8xx+x)dM

dx=1

x+x+x(1

2+1

4x+x(2x

2x)+1

8xx+x)dM

dx=1

x+x+x(1

2+2x+1

8xx+x)dM

dx=1

x+x+x(1

2(4xx+x

4xx+x)+2x+1

8xx+x)dM

dx=4xx+x)+2x+1

x+x+x 3) f(x)=a

x3sin(1 x4)		se x≠0
0		se x=0

df

dx=d

dx(x3)·sin(1

x4)+x3·d

dx(sin(1

x4)d

dx(1

x4))df

dx=3x2sin(1

x4)+x3cos(1

x4)(-4x-5)df

dx=3x2sin(1

x4)-4

x2cos(1

x4)df

dx=df

dx(0)=0 Gilson Würz in Math