Lösungen

1

a

D[ a4 x^4 + 2 x^3+ a2 x^2 + 5 x - 6, x]

b

D[Cos[x]-E^x+1/x,x]

c

D[Sin[x] Sqrt[x]-Log[Pi x],x]

d

D[Sin[x] E^(-x)-x/Log[x],x]

e

D[Cos[2+ E^x] - E^(-x^2),x]

f

D[x^x,x]

D[E^(Log[x] x),x]

2

a

Plot[ 3 x^2 - 4 2 x + 1,{x,-2,5}];

b

sol1=Solve[ 3 x^2 - 4 α x  +1 == 0,{x}]//Flatten

x1=x/.sol1[[1]]

x2=x/.sol1[[2]]

D[3 x^2-4 α x+1,x]==1

q1=Evaluate[D[3 x^2-4 α x+1,x]==1/.x->x1]//Simplify

q2=Evaluate[D[3 x^2-4 α x+1,x]==1/.x->x2]//Simplify

Solve[q2,{α}]

N[%]

c

D[x (3 x^2-4 α x+1),{x,2}]

gleich = (D[x (3 x^2-4 α x+1),{x,2}]==0/.x->1/2)  //Simplify

Solve[Evaluate[gleich],{α}]//Flatten

N[%]

3

a

Limit[1/Sin[x]-1/x,x->0]

b

Limit[(x-1)(Log[x-1])+(x^2-1)/(x^3-1),x->1]

4

Plot[Abs[(x^2-9)(x+3)],{x,-6,4}];

Extrema bei Nullstellen sowie dort wo die Ableitung Null ist. Ebenso am Rande (ablesbar am Intervall).

Solve[(x^2-9)(x+3)==0,{x}]//Flatten

Solve[Evaluate[D[(x^2-9)(x+3),x]==0],{x}]//Flatten

5

Plot[(Sinh[x-3]-10)^2,{x,-1,8}];

Plot[(Sinh[x-3]-10)^2,{x,5.99,6.005}];

Solve[Evaluate[D[(Sinh[x-3]-10)^2,x]==0],{x}]//Flatten//N

Extrema:  Gültiger Wert:   x -> 5.99822.  Ebenso Randestrema (ablesbar am Intervall).

Exakte Rechnung (Achtung, sehr grosser Output!):

Solve[Evaluate[D[(Sinh[x-3]-10)^2,x]==0],{x}]//Flatten//N

Solve[Evaluate[D[(Sinh[x-3]-10)^2,{x,2}]==0],{x}]//Flatten//N//Chop

Plot[(Sinh[x-3]-10)^2,{x, 2.2, 5.9}];

Wendepunkte:  x->0.707781, x->5.29222

Solve[Evaluate[D[(Sinh[x-3]-10)^2,{x,2}]==0],{x}]//Flatten//Simplify

6

Plot[ArcSin[Cos[x]], {x, -10, 10}, 
        AspectRatio -> Automatic];

Plot[Evaluate[D[ArcSin[Cos[x]], x]], {x, -10, 10}, 
        AspectRatio -> Automatic];