Funktion derivaatta

f(x)=x^2
f(x)=x2f(x)=x^2

 kulmakerroin on funktion f derivaatta kohdassa 1.

Funktion f  kuvaajalle

 muuttujan arvon 1 kohdalle

 piirretyn tangentin

f'(1)=2
f′(1)=2f'(1)=2
k_t=\dfrac{4}{2}= 2
kt=42=2k_t=\dfrac{4}{2}= 2
2
22
4
44

Derivaatta

Liikuta punaista pistettä ja keksi sääntö derivaatan laskemiselle

Derivaatta

x f ' (x)
-2 -4
-1 2
0 0
1 2
2 4
f'(x)=2x
f′(x)=2xf'(x)=2x
2 \cdot (-2)=-4
2⋅(−2)=−42 \cdot (-2)=-4
2 \cdot (-1)=-2
2⋅(−1)=−22 \cdot (-1)=-2
2 \cdot 0=0
2⋅0=02 \cdot 0=0
2 \cdot 2=4
2⋅2=42 \cdot 2=4
2 \cdot 1 = 2
2⋅1=22 \cdot 1 = 2

Pohdintaa...

f(x)=x^2
f(x)=x2f(x)=x^2

Tangentit

Graafinen derivointi

tangentin kulmakertoimen määrittäminen

Algebrallinen derivointi

siirrytään

derivoimissäännöt

f(x)=x^2
f(x)=x2f(x)=x^2
f'(x)=2x
f′(x)=2xf'(x)=2x

Derivoimissääntöjä

Dx^\color{Salmon} n=\color{Salmon}nx^{\color{Salmon}n-1}
Dxn=nxn−1Dx^\color{Salmon} n=\color{Salmon}nx^{\color{Salmon}n-1}
D\color{CornflowerBlue} kx=\color{CornflowerBlue}k
Dkx=kD\color{CornflowerBlue} kx=\color{CornflowerBlue}k
D\color{Yellow} k=0
Dk=0D\color{Yellow} k=0
D(\color{Yellow} 5)=0
D(5)=0D(\color{Yellow} 5)=0
D(\color{Yellow} 2)=0
D(2)=0D(\color{Yellow} 2)=0
D(\color{Yellow} -\color{Yellow} 1\color{Yellow} 0)=0
D(−10)=0D(\color{Yellow} -\color{Yellow} 1\color{Yellow} 0)=0
D(\color{CornflowerBlue} 5x)=\color{CornflowerBlue} 5
D(5x)=5D(\color{CornflowerBlue} 5x)=\color{CornflowerBlue} 5
D(\color{CornflowerBlue} -x)=\color{CornflowerBlue} {-1}
D(−x)=−1D(\color{CornflowerBlue} -x)=\color{CornflowerBlue} {-1}
D(\color{CornflowerBlue} 2\color{CornflowerBlue} 8x)=\color{CornflowerBlue} 2\color{CornflowerBlue} 8
D(28x)=28D(\color{CornflowerBlue} 2\color{CornflowerBlue} 8x)=\color{CornflowerBlue} 2\color{CornflowerBlue} 8
D(x^\color{Salmon} 2)=\color{Salmon} 2x^{\color{Salmon}2-1}=\color{Salmon} 2x
D(x2)=2x2−1=2xD(x^\color{Salmon} 2)=\color{Salmon} 2x^{\color{Salmon}2-1}=\color{Salmon} 2x
D(x^\color{Salmon} 3)=\color{Salmon} 3x^{\color{Salmon}3-1}=\color{Salmon} 3x^2
D(x3)=3x3−1=3x2D(x^\color{Salmon} 3)=\color{Salmon} 3x^{\color{Salmon}3-1}=\color{Salmon} 3x^2
D(x^\color{Salmon}8)=\color{Salmon} 8x^{\color{Salmon}8-1}=\color{Salmon} 8x^7
D(x8)=8x8−1=8x7D(x^\color{Salmon}8)=\color{Salmon} 8x^{\color{Salmon}8-1}=\color{Salmon} 8x^7

Esimerkki

Esimerkki

D(3x^\color{Salmon}2+\color{Salmon} 4x+\color{Salmon} 5)=3 \cdot \color{Salmon} 2x^{\color{Salmon}2-1}+\color{CornflowerBlue} 4+0=6x+4
D(3x2+4x+5)=3⋅2x2−1+4+0=6x+4D(3x^\color{Salmon}2+\color{Salmon} 4x+\color{Salmon} 5)=3 \cdot \color{Salmon} 2x^{\color{Salmon}2-1}+\color{CornflowerBlue} 4+0=6x+4