Logaritmin derivaatta

Logaritmin derivaatta https://peda.net/id/0d00dda271e 2025-08-05T13:50:46+03:00 https://peda.net/:static/539/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> Teoria ja esimerkit https://peda.net/id/0d0258e571e 2018-03-14T09:13:33+02:00 Logaritmin derivaatta<br/> <ul> <li><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D%5Cleft%28%5Cln%20x%5Cright%29%3D%5Cfrac%7B1%7D%7Bx%7D%7B%2C%7D%5C%20%5C%20%5C%20%5Ctext%7Bkun%7D%5C%20x%3E0" alt="\text{D}\left(\ln x\right)=\frac{1}{x}{,}\ \ \ \text{kun}\ x>0"/></li> <li><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D%5Cleft%28%5Cln%5Cright%7Cx%5Cleft%7C%5Cright%29%3D%5Cfrac%7B1%7D%7Bx%7D%7B%2C%7D%5C%20%5C%20%5C%20%5Ctext%7Bkun%7D%5C%20x%5Cne0" alt="\text{D}\left(\ln\right|x\left|\right)=\frac{1}{x}{,}\ \ \ \text{kun}\ x\ne0"/><br/> <br/> </li> <li><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D%5Cleft(%5Clog_ax%5Cright)%3D%5Cfrac%7B1%7D%7Bx%5Cln%20a%7D%7B%2C%7D%5C%20%5C%20%5C%20%5Ctext%7Bkun%7D%5C%20x%3E0" alt="\text{D}\left(\log_ax\right)=\frac{1}{x\ln a}{,}\ \ \ \text{kun}\ x>0"/></li> </ul> <br/> <b>ESIM 1<br/> </b>Derivoi<br/> a) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln2x" alt="\ln2x"/><br/> b) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5C%20x%5E2" alt="\ln\ x^2"/><br/> c) <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cln%20x" alt="x\ln x"/><br/> d) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(2x%5E5-4x%5E2%5Cright)" alt="\ln\left(2x^5-4x^2\right)"/><br/> <br/> <b>ESIM 2</b><br/> Mikä on funktion <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft%28x%5Cright%29%3D%5Cfrac%7B%5Cln%20x%7D%7Bx%7D" alt="f\left(x\right)=\frac{\ln x}{x}"/>, x > 0, suurin arvo? [K12/5]<br/> <br/> 2025-08-05T13:50:46+03:00 Vastaukset https://peda.net/id/0d02aeea71e 2018-03-14T14:41:18+02:00 <b>ESIM 1</b><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D%5Cln2x%3D%5Cfrac%7B1%7D%7B2x%7D%5Ccdot2%3D%5Cfrac%7B1%7D%7Bx%7D" alt="\text{D}\ln2x=\frac{1}{2x}\cdot2=\frac{1}{x}"/>, x > 0<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D%5Cln%20x%5E2%3D%5Ctext%7BD%7D2%5Cln%20x%3D2%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%3D%5Cfrac%7B2%7D%7Bx%7D" alt="\text{D}\ln x^2=\text{D}2\ln x=2\cdot\frac{1}{x}=\frac{2}{x}"/>, x > 0<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D%5Cleft%28x%5Cln%20x%5C%20%5Cright%29%3D1%5Ccdot%5Cln%20x%2Bx%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%3D%5Cln%20x%2B1" alt="\text{D}\left(x\ln x\ \right)=1\cdot\ln x+x\cdot\frac{1}{x}=\ln x+1"/>, x > 0<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D%5Cleft%28%5Cln%5Cleft%282x%5E5%2B4x%5E2%5Cright%29%5Cright%29%3D%5Cfrac%7B1%7D%7B2x%5E5%2B4x%5E2%7D%5Ccdot%5Cleft%2810x%5E4%2B8x%5Cright%29%3D%5Cfrac%7B2x%5Cleft%285x%5E3%2B4%5Cright%29%7D%7B2x%5Cleft%28x%5E4%2Bx%5Cright%29%7D%3D%5Cfrac%7B5x%5E3%2B4%7D%7Bx%5E4%2Bx%7D" alt="\text{D}\left(\ln\left(2x^5+4x^2\right)\right)=\frac{1}{2x^5+4x^2}\cdot\left(10x^4+8x\right)=\frac{2x\left(5x^3+4\right)}{2x\left(x^4+x\right)}=\frac{5x^3+4}{x^4+x}"/>, x > 0<br/> <br/> <br/> <b>ESIM 2</b><br/> <div> <div> <div>Mikä on funktion <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft%28x%5Cright%29%3D%5Cfrac%7B%5Cln%20x%7D%7Bx%7D" alt="f\left(x\right)=\frac{\ln x}{x}"/>, x > 0, suurin arvo? [K12/5] </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cfrac%7Bf%27g-fg%27%7D%7Bg%5E2%7D%3D%5Cfrac%7B%5Cfrac%7B1%7D%7Bx%7D%5Ccdot%20x-%5Cln%20x%5Ccdot1%7D%7Bx%5E2%7D%3D%5Cfrac%7B1-%5Cln%20x%7D%7Bx%5E2%7D" alt="f'\left(x\right)=\frac{f'g-fg'}{g^2}=\frac{\frac{1}{x}\cdot x-\ln x\cdot1}{x^2}=\frac{1-\ln x}{x^2}"/></div> <div>Etsitään nollakohdat:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D0" alt="f'\left(x\right)=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1-%5Cln%20x%7D%7Bx%5E2%7D%3D0" alt="\frac{1-\ln x}{x^2}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1-%5Cln%20x%3D0" alt="1-\ln x=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20x%3D1" alt="\ln x=1"/><span> || </span><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B%5Cleft(%5C%20%5Cright)%7D" alt="e^{\left(\ \right)}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3De%5E1%3De" alt="x=e^1=e"/><br/> <div>Kuuluu määrittelyjoukkoon.</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(1%5Cright)%3D%5Cfrac%7B1-%5Cln1%7D%7B1%5E2%7D%3D%5Cfrac%7B1-0%7D%7B1%7D%3D1%3E0" alt="f'\left(1\right)=\frac{1-\ln1}{1^2}=\frac{1-0}{1}=1>0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(3%5Cright)%3D%5Cfrac%7B1-%5Cln3%7D%7B3%5E2%7D%3C0%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(%5Cln3%3E1%5Cright)" alt="f'\left(3\right)=\frac{1-\ln3}{3^2}<0\ \ \ \ \ \ \ \ \ \ \ \left(\ln3>1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%260%26%26e%26%5C%5C%0A%5Chline%0Af%27%5Cleft(x%5Cright)%26%5Cmathrm%7Bei%5C%20m%C3%A4%C3%A4r.%7D%26%2B%260%26-%5C%5C%0Af%5Cleft(x%5Cright)%26%5Cmathrm%7Bei%5C%20m%C3%A4%C3%A4r.%7D%26%5Cnearrow%26%5Cmathrm%7Bmaks%7D%26%5Csearrow%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}
&0&&e&\\
\hline
f'\left(x\right)&\mathrm{ei\ määr.}&+&0&-\\
f\left(x\right)&\mathrm{ei\ määr.}&\nearrow&\mathrm{maks}&\searrow
\end{array}"/> <div>Kulkukaavion perusteella suurin arvo on <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(e%5Cright)%3D%5Cfrac%7B%5Cln%20e%7D%7Be%7D%3D%5Cfrac%7B1%7D%7Be%7D" alt="f\left(e\right)=\frac{\ln e}{e}=\frac{1}{e}"/></div> </div> 2025-08-05T13:50:46+03:00