3.2 Neperin luku ja eksponenttifunktion derivaatta

328

a)

$\pi>e>\frac{1}{2}>\frac{1}{e}>\frac{1}{3}>\frac{1}{\pi}$

b)

$e-1>\frac{e}{3}$

$e-1\approx1{,}72$
$\frac{e}{3}\approx0{,}91$

327

a)
$f\left(x\right)=\frac{1}{2}e^x\left(\sin x+\cos x\right)$
$f'\left(x\right)=\frac{1}{2}e^x\left(\cos x-\sin x\right)$

$f'\left(0\right)=\frac{1}{2}\cdot1\cdot\left(1-0\right)=\frac{1}{2}$

b)

$f\left(x\right)=x^3e^x$
$f'\left(x\right)=x^3e^x+3x^2e^x$
$f'\left(x\right)=0$
$x=-3\ tai\ x=0$

326

a)

$D\left(e^{x^3-1}\right)=3x^2e^{x^3-1}$

b)

$D\ \frac{e^x}{x}=\frac{e^x\cdot x-1\cdot e^x}{x^2}=\frac{e^x\left(x-1\right)}{x^2}$

c)

$D\ \frac{x+1}{e^x}=\frac{1\cdot e^{2x}-2e^{2x}\cdot\left(x+1\right)}{e^{2x}}==-x+e^{-x}$

d)

$D\frac{2}{e^{2x}}=2e^{-2x}=-4e^{-2x}$

e)

$D\left(x^2e^{2x}\right)=2x\cdot e^{2x}+x^2\cdot e^{2x}=e^{2x}\left(x^2+2x\right)$

f)

$D\left(\frac{e^x+x}{x}\right)=\frac{x\left(e^x+1\right)-1\left(e^x+x\right)}{x^2}=\frac{\left(x-1\right)\left(e^x+x\right)}{x^2}$

321

a)

$D\left(2e^x+2x\right)=2e^x+2$

b)

$D\left(e^{3x}+e\right)=3xe^{3x}$

c)

$D\left(\frac{e^{6x}}{3}\right)=\frac{6e^{6x}\cdot3-e^{6x}\cdot0}{3^2}=\frac{18e^{6x}}{9}=2e^{6x}$

d)

$D\left(xe^x\right)=1\cdot e^x+x\cdot e^x=\left(1+x\right)e^x$

323

a)
$f(0)=1$
$f(1)=e$
b)
$e^x=1$
$x=0$
c)
$e^{-1}$