ableitungen-üben

3.28a)

$$||"id":1609097473726||f(x)=2x^2+2x+1f^{\prime }(x)=4x^1+2$$

c)

$$||"id":50981288364||f(x)=\frac{3}{x^2}-\frac{4}{x}=3x^{-2}-4x^{-1}{f}^{\prime }(x)=-6x^{-3}+4x^{-2}=-\frac{6}{x^3}+\frac{4}{x^2}$$

f)

$$||"id":249053329159||f(x)=\frac{x^2+1}{2x}=\frac{x^2}{2x}+\frac{1}{2x}=\frac{x}{2}+\frac{x^{-1}}{2}{f}^{\prime }(x)=\frac{1}{2}-\frac{1x^{-2}}{2}=-\frac{1}{2x^2}+\frac{1}{2}$$

3.26) a)

$$||"id":1454345359785||f(x)=2x^2{f}^{\prime }(x)=2\ast 2x^{2-1}=4x$$

b)

$$||"id":877527017994||f(x)=\frac{x}{2}{f}^{\prime }(x)=\frac{1\ast x^{1-1}}{2}=\frac{1\ast x^0}{2}=\frac{1}{2}$$

c)

$$||"id":970221368795||f(x)=2x^{-1}{f}^{\prime }(x)=-1\ast 2x^{-1-1}=-1\ast 2x^{-2}=-2x^{-2}=\frac{.2}{x^2}$$

d)

$$||"id":757498232671||f(x)=\frac{1}{3x}=\frac{x^{-1}}{3}{f}^{\prime }(x)=\frac{-1x^{-2}}{3}=-\frac{1}{3x^2}$$

e)

$$||"id":1352343433582||f(x)=\frac{1}{x^2}=x^{-2}{f}^{\prime }(x)=-2x^{-3}=\frac{2}{x^3}$$

f)

$$||"id":164478647853||f(x)=\frac{1}{4x^3}=\frac{x^{-3}}{4}{f}^{\prime }(x)=\frac{3}{4}\ast x^{-4}=-\frac{3}{4x^4}$$

g)

$$||"id":146415033688||f(x)=(3x{)}^2=9x^2{f}^{\prime }(x)=18x$$

h)

$$||"id":1209908531270||f(x)=\sqrt[3]{x}=x^{\frac{1}{3}}{f}^{\prime }(x)=\frac{1}{3}\ast x^{-\frac{2}{1}}=\frac{1}{3\ast \sqrt[3]{x^2}}$$

i)

$$||"id":1062332842432||f(x)=x\ast \sqrt{x}=x^{\frac{2}{2}}{f}^{\prime }(x)=\frac{3}{2}\ast x^{1/2-2/2}=\frac{3}{2}\sqrt{x}$$

j)

$$||"id":104414381423||f(x)=\sqrt{2x}=\sqrt{2}\ast x^{\frac{1}{2}}{f}^{\prime }(x)=\frac{\sqrt{2}}{2\ast \sqrt{x}}=\frac{\sqrt{2}}{\sqrt{2^{2}x}}=\sqrt{\frac{2}{2^2\times }}=\sqrt{\frac{1}{2x}}=\frac{1}{\sqrt{2x}}$$

3.32)a)

$$||"id":1424555366248||y=2x^2-12x+21y^{\prime }=4x-120=4x-1212=4xx=32\ast 3^2-12\ast 3+21=3S(3|3)$$

3.35)

$$||"id":295119000421||y=a{x}^2+bx+c{y}^{\prime }=2ax+bx=3,Steigung0.5\Rightarrow f^{\prime }(3)=0.5\Rightarrow 2\ast a\ast 3+b=0.5$$

P(1|3),x=0⇒y-Achse geschnitten

$$||"id":734708710342||f(1)=00=a\ast 1^2+b\ast 1+c$$

P(0|3),x=0⇒y-Achse geschnitten

$$||"id":1532632945501||3=a\ast 0+b\ast 0+c\Rightarrow c=3$$ $$||"id":289643842388|||:-0.5=6a+b|-||:0=a+b+3|-0.5=5a-32.5=5aa=\frac{1}{2}0=\frac{1}{2}+b+3b=-\frac{7}{2}\Rightarrow y=\frac{1}{2}{x}^2-\frac{7}{2}x+3$$

3.42) a)

$$||"id":839691508885||s(t)=v_{0}t+\frac{1}{2}a\ast t^2{s}^{\prime }(t)=v(t)v^{\prime }(t)=a(t)s^{\prime }(t)=v(t)=v_0+a\ast t$$

b)

$$||"id":1385642251437||v_0=45kmha=4\frac{m}{s^2}v(3)=\frac{45}{3.6}+4\ast 3=24m/s=88,2kmh$$

3.28) b)

$$||"id":1181985723778||f(x)=x+x^{-1}{f}^{\prime }(x)=1-1\ast x^{-2}=1-x^{-2}=1-\frac{1}{x^{-2}}$$

d)

$$||"id":658600491113||f(x)=\frac{x+1}{2}=\frac{x}{2}+\frac{1}{2}{f}^{\prime }(x)=\frac{1}{2}$$

e)

$$||"id":1655073174483||f(x)=\frac{x+1}{x}=\frac{x}{x}+\frac{1}{x}=1+1x^{-1}{f}^{\prime }(x)=-x^{-2}=-\frac{1}{x^2}$$

3.29) a)

$$||"id":1265166888200||f(x)=-\cos x+3\ast \tan x{f}^{\prime }(x)=\sin x+\frac{3}{{\cos }^{2}x}$$

e)

$$||"id":153521589843||f(x)=\ln x+{\ln }_2{f}^{\prime }(x)=\frac{1}{x}$$

b)

$$||"id":112495509974||f(x)=\frac{\sin x}{2\ast \cos x}=\frac{1}{2}\ast \tan x{f}^{\prime }(x)=\frac{1}{2\ast {\cos }^{2}x}$$

c)

$$||"id":83297599398||f(x)=\frac{x}{2}+e^x{f}^{\prime }(x)=\frac{1}{2}+e^x$$

d)

$$||"id":526844798184||f(x)=\frac{3x^2-2^x}{2}=\frac{3}{2}\ast x^2-\frac{1}{2}\ast 2^x{f}^{\prime }(x)=3x-\frac{2^x\ast \ln 2}{2}$$

f)

$$||"id":876959150763||f(x)=x+3lgx{f}^{\prime }(x)=\frac{1}{2}+e^x$$ $$||"id":1060471443456||f(x)=3x+x^2{f}^{\prime }(x)=3+2xf(x)=(3x+x^2)\ast (x-1)=3x^2+x^3-3x-x^2=x^3+2x-3x{f}^{\prime }(x)=3x^2+4x-3$$