Derivoi.
a) \(-8\)
b) $2,\!9x$
c) $-\frac{2}{3}x$
d) $-\frac{x}{4}$
e) $x^5$
f) $-1,\!8x^4$
a) D$(-8)=0$ $\color{Red}{(+1\text{p})}$
b) D$(2,9x)=2,9$ $\color{Red}{(+1\text{p})}$
c) D$(-\frac{2}{3}x)=-\frac{2}{3}$ $\color{Red}{(+1\text{p})}$
d) D$(-\frac{x}{4})=-\frac{1}{4}$ $\color{Red}{(+1\text{p})}$
e) D$(x^5)=5x^4$ $\color{Red}{(+1\text{p})}$
f) D$(-1,8x^4)=-1,8\cdot 4x^3=-7,2x^3$ $\color{Red}{(+1\text{p})}$
Derivoi funktiot.
a) $f(x)=\frac{3}{5}x^4+2x^2+7x-9$
b) $h(t)=\frac{2t+6}{8}$
c) $m(r)=5(r^3+2r^2-4r+8)$
a)
\(\begin{align} f'(x)&=D(\frac{3}{5}x^4+2x^2+7x-9)\\ &=\frac{3}{5}\cdot 4x^3+2\cdot 2x+7\cdot 1-0 &\color{Red}{(+1\text{p})}\\ &=\frac{12}{5}x^3+4x+7 &\color{Red}{(+1\text{p})} \end{align}\)
b)
\(\begin{align} h'(t)&=D(\frac{2t+6}{8}) \\ &= D(\frac{2t}{8}+\frac{6}{8}) &\color{Red}{(+1\text{p})}\\ &= \frac{2}{8}+0&\color{Red}{(+\frac{1}{2}\text{p})}\\ &=\frac{1}{4}& \color{Red}{(+\frac{1}{2}\text{p})} \end{align}\)
c)
\(\begin{align} m'(r)&=D(5(r^3+2r^2-4r+8))\\ &=D(5r^3+10r^2-20r+40)&\color{Red}{(+1\text{p})}\\ &=5\cdot3r^2+10\cdot 2r-20 \cdot 1+0 &\color{Red}{(+\frac{1}{2}\text{p})}\\ &=15r^2+20r-20 &\color{Red}{(+\frac{1}{2}\text{p})} \end{align}\)
Olkoon $f(x)=-3x^2+12x-15$.
a) Määritä funktion $f$ derivaatan nollakohdat.
b) Määritä $f'(-2)$.
c) Ratkaise yhtälö $f'(x)=-12$.
\(\begin{align} f'(x)&=D(-3x^2+12x-15) \\ &=-3\cdot 2x+12 \cdot 1 - 0 &\color{Red}{(+\frac{1}{2}\text{p})} \\ &=-6x+12 &\color{Red}{(+\frac{1}{2}\text{p})} \end{align}\)
a) Ratkaistaan funktion $f$ derivaatan nollakohdat:
\(\begin{align} f'(x)&=0 &\color{Red}{(+\frac{1}{2}\text{p})}\\ -6x+12 &= 0 &\color{Red}{(+\frac{1}{2}\text{p})}\\ -6x &= -12 \quad ||:(-6) \\ x&=2 \end{align}\)
Vastaus: $x=2$ $\color{Red}{(+1\text{p})}$
b) Määritetään $f'(-2)$:
\(\begin{align} f'(-2)&=-6\cdot (-2)+12 &\color{Red}{(+\frac{1}{2}\text{p})} \\ &=12+12 \\ &= 24 \end{align}\).
Vastaus: $f'(-2)=24$ $\color{Red}{(+1\text{p})}$
c) Ratkaistaan yhtälö
\(\begin{align} f'(x)&=-12 \\ -6x+12&=-12 &\color{Red}{(+\frac{1}{2}\text{p})}\\ -6x&= -12-12 \\ -6x&=-24 \quad ||:(-6) \\ x&=4 \end{align}\)
Vastaus: $f'(4)=12$ $\color{Red}{(+1\text{p})}$
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