https://peda.net/id/e26dd688f97 2019-10-28T12:51:24+02:00 https://peda.net/:static/537/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/95426e4af97 2019-10-28T14:07:59+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B1%7D%7Bx%5E2%7D%3D" alt="D\ \frac{1}{x^2}="/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D1%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D0%5C%20ja%5C%20g%5Cleft(x%5Cright)%3Dx%5E2%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D2x" alt="f\left(x\right)=1{,}\ f'\left(x\right)=0\ ja\ g\left(x\right)=x^2{,}\ g'\left(x\right)=2x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B0%5Cleft(x%5E2%5Cright)-2x%7D%7Bx%5E4%7D%3D%5Cfrac%7B-2x%7D%7Bx%5E4%7D%3D%5Cfrac%7B-2%7D%7Bx%5E3%7D" alt="\frac{0\left(x^2\right)-2x}{x^4}=\frac{-2x}{x^4}=\frac{-2}{x^3}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B1%7D%7Bx%5E2%7D%3D1%5Ccdot%20x%5E%7B-2%7D%3D-2x%5E%7B-3%7D%3D%5Cfrac%7B-2%7D%7Bx%5E%7B-3%7D%7D" alt="D\ \frac{1}{x^2}=1\cdot x^{-2}=-2x^{-3}=\frac{-2}{x^{-3}}"/></div> 2019-10-28T14:07:59+02:00 https://peda.net/id/f8f45328f97 2019-10-28T14:04:35+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7B2%7D%7B2x%2B4%7D%5Cright)%3D" alt="D\left(\frac{2}{2x+4}\right)="/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D0%5C%20ja%5C%20g%5Cleft(x%5Cright)%3D2x%2B4%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D2" alt="f\left(x\right)=2{,}\ f'\left(x\right)=0\ ja\ g\left(x\right)=2x+4{,}\ g'\left(x\right)=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B0%5Cleft(2x%2B4%5Cright)-2%5Ccdot2%7D%7B%5Cleft(2x%2B4%5Cright)%5E2%7D%3D-%5Cfrac%7B4%7D%7B%5Cleft(2x%2B4%5Cright)%5E2%7D%3D-%5Cfrac%7B1%7D%7B%5Cleft(x%2B1%5Cright)%5E2%7D" alt="\frac{0\left(2x+4\right)-2\cdot2}{\left(2x+4\right)^2}=-\frac{4}{\left(2x+4\right)^2}=-\frac{1}{\left(x+1\right)^2}"/> <br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7B2x%2B4%7D%7B2%7D%5Cright)%3D" alt="D\left(\frac{2x+4}{2}\right)="/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%2B4%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D2%5C%20ja%5C%20g%5Cleft(x%5Cright)%3D2%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D0" alt="f\left(x\right)=2x+4{,}\ f'\left(x\right)=2\ ja\ g\left(x\right)=2{,}\ g'\left(x\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%5Ccdot2-0%5Cleft(2x%2B4%5Cright)%7D%7B2%5E2%7D%3D%5Cfrac%7B4%7D%7B4%7D%3D1" alt="\frac{2\cdot2-0\left(2x+4\right)}{2^2}=\frac{4}{4}=1"/> 2019-10-28T14:03:36+02:00 https://peda.net/id/1060a1e8f97 2019-10-28T13:57:06+02:00 a) <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7B3x%7D%7Bx%2B1%7D%5Cright)%3D" alt="D\left(\frac{3x}{x+1}\right)="/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3x%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D3%5C%20ja%5C%20g%5Cleft(x%5Cright)%3Dx%2B1%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D1" alt="f\left(x\right)=3x{,}\ f'\left(x\right)=3\ ja\ g\left(x\right)=x+1{,}\ g'\left(x\right)=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%5Cleft(x%2B1%5Cright)-3x%5Ccdot1%7D%7B%5Cleft(x%2B1%5Cright)%5E2%7D%3D%5Cfrac%7B3x%2B3-3x%7D%7B%5Cleft(x%2B1%5Cright)%5E2%7D%3D%5Cfrac%7B3%7D%7B%5Cleft(x%2B1%5Cright)%5E2%7D" alt="\frac{3\left(x+1\right)-3x\cdot1}{\left(x+1\right)^2}=\frac{3x+3-3x}{\left(x+1\right)^2}=\frac{3}{\left(x+1\right)^2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7Bx%5E2%7D%7B2x%2B1%7D%5Cright)" alt="D\left(\frac{x^2}{2x+1}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D2x%5C%20ja%5C%20g%5Cleft(x%5Cright)%3D2x%2B1%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D2" alt="f\left(x\right)=x^2{,}\ f'\left(x\right)=2x\ ja\ g\left(x\right)=2x+1{,}\ g'\left(x\right)=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x%5Cleft(2x%2B1%5Cright)-2x%5E2%7D%7B%5Cleft(2x%2B1%5Cright)%5E2%7D%3D%5Cfrac%7B4x%5E2%2B2x-2x%5E2%7D%7B%5Cleft(2x%2B1%5Cright)%5E2%7D%3D%5Cfrac%7B2x%5E2%2B2x%7D%7B%5Cleft(2x%2B1%5Cright)%5E2%7D" alt="\frac{2x\left(2x+1\right)-2x^2}{\left(2x+1\right)^2}=\frac{4x^2+2x-2x^2}{\left(2x+1\right)^2}=\frac{2x^2+2x}{\left(2x+1\right)^2}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7Bx%7D%7B1-x%5E2%7D%5Cright)%3D" alt="D\left(\frac{x}{1-x^2}\right)="/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D1%5C%20ja%5C%20g%5Cleft(x%5Cright)%3D1-x%5E2%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D-2x" alt="f\left(x\right)=x{,}\ f'\left(x\right)=1\ ja\ g\left(x\right)=1-x^2{,}\ g'\left(x\right)=-2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E2%2B1%7D%7B%5Cleft(1-x%5E2%5Cright)%5E2%7D" alt="\frac{x^2+1}{\left(1-x^2\right)^2}"/></div> 2019-10-28T13:57:06+02:00 https://peda.net/id/eb7da080f97 2019-10-28T13:41:45+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(x%5E2%2B1%5Cright)%5Cleft(x%5E2-1%5Cright)" alt="D\left(x^2+1\right)\left(x^2-1\right)"/><br/> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%2B1%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D2x%5C%20ja%5C%20g%5Cleft(x%5Cright)%3Dx%5E2-1%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D2x" alt="f\left(x\right)=x^2+1{,}\ f'\left(x\right)=2x\ ja\ g\left(x\right)=x^2-1{,}\ g'\left(x\right)=2x"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(f%5Cleft(x%5Cright)%5Cleft(g%5Cleft(x%5Cright)%5Cright)%5Cright)%3Df%5Cleft(x%5Cright)g'%5Cleft(x%5Cright)%2Bf'%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%3D%5Cleft(x%5E2%2B1%5Cright)%5Ccdot2x%2B2x%5Cleft(x%5E2-1%5Cright)" alt="D\left(f\left(x\right)\left(g\left(x\right)\right)\right)=f\left(x\right)g'\left(x\right)+f'\left(x\right)g\left(x\right)=\left(x^2+1\right)\cdot2x+2x\left(x^2-1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D2x%5E3%2B2x%2B2x%5E3-2x%3D4x%5E3" alt="=2x^3+2x+2x^3-2x=4x^3"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%5E2%2B1%5Cright)%5Cleft(x%5E2-1%5Cright)%3Dx%5E4-x%5E2%2Bx%5E2-1%3Dx%5E4-1%3D4x%5E3" alt="\left(x^2+1\right)\left(x^2-1\right)=x^4-x^2+x^2-1=x^4-1=4x^3"/></div> 2019-10-28T13:41:45+02:00 https://peda.net/id/5d331a08f97 2019-10-28T13:37:47+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7B1%7D%7Bx%5E2%7D%5Cright)%3Dx%5E%7B-2%7D" alt="D\left(\frac{1}{x^2}\right)=x^{-2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7B4%7D%7Bx%5E3%7D%5Cright)%3D4%5Ccdot%5Cfrac%7B1%7D%7Bx%5E3%7D%3D4x%5E%7B-3%7D%3D12x%5E%7B-4%7D%3D%5Cfrac%7B12%7D%7Bx%5E4%7D" alt="D\left(\frac{4}{x^3}\right)=4\cdot\frac{1}{x^3}=4x^{-3}=12x^{-4}=\frac{12}{x^4}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(-%5Cfrac%7B6%7D%7Bx%5E7%7D%5Cright)%3D-6x%5E%7B-7%7D%3D-42x%5E%7B-8%7D%3D-%5Cfrac%7B42%7D%7Bx%5E%7B-8%7D%7D" alt="D\left(-\frac{6}{x^7}\right)=-6x^{-7}=-42x^{-8}=-\frac{42}{x^{-8}}"/></div> 2019-10-28T13:37:47+02:00 https://peda.net/id/898664d2f97 2019-10-28T13:17:33+02:00 <div>Lause</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Dx%5En%3Dnx%5E%7Bn-1%7D%7B%2C%7D%5C%20kun%5C%20x%5Cne0%5C%20ja%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="Dx^n=nx^{n-1}{,}\ kun\ x\ne0\ ja\ n\in\mathbb{Z}"/></div> <div> </div> <div>Muista!</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E%7B-k%7D%3D%5Cfrac%7B1%7D%7Bx%5Ek%7D" alt="x^{-k}=\frac{1}{x^k}"/></div> <div> </div> <div>Esimerkki. Derivoi</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%7D%7Bx%5E3%7D%7B%2C%7D%5C%20x%5Cne0" alt="\frac{3}{x^3}{,}\ x\ne0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B3%7D%7Bx%5E3%7D%3DD%5C%203%5Ccdot%5Cfrac%7B1%7D%7Bx%5E3%7D%3D3x%5E%7B-3%7D%3D3%5Cleft(-3%5Cright)x%5E%7B-3-1%7D%3D-9x%5E%7B-4%7D%3D-%5Cfrac%7B9%7D%7Bx%5E4%7D" alt="D\ \frac{3}{x^3}=D\ 3\cdot\frac{1}{x^3}=3x^{-3}=3\left(-3\right)x^{-3-1}=-9x^{-4}=-\frac{9}{x^4}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E4-2%7D%7Bx%5E4%7D%7B%2C%7Dx%5Cne0%5C%20%5C%20D%5Cleft(%5Cfrac%7Bx%5E4%7D%7Bx%5E4%7D-%5Cfrac%7B2%7D%7Bx%5E4%7D%5Cright)%3DD%5Cleft(1-%5Cfrac%7B2%7D%7Bx%5E4%7D%5Cright)" alt="\frac{x^4-2}{x^4}{,}x\ne0\ \ D\left(\frac{x^4}{x^4}-\frac{2}{x^4}\right)=D\left(1-\frac{2}{x^4}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3DD%5Cleft(1-2x%5E%7B-4%7D%5Cright)%3D8x%5E%7B-5%7D%3D%5Cfrac%7B8%7D%7Bx%5E%7B-5%7D%7D" alt="=D\left(1-2x^{-4}\right)=8x^{-5}=\frac{8}{x^{-5}}"/></div> <div> </div> <div>Lause</div> <div>Olkoon f ja g derivoituvia. Tällöin</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(f%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%5Cright)%3Df'%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%2Bf%5Cleft(x%5Cright)g'%5Cleft(x%5Cright)" alt="D\left(f\left(x\right)g\left(x\right)\right)=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7Bf%5Cleft(x%5Cright)%7D%7Bg%5Cleft(x%5Cright)%7D%5Cright)%3D%5Cfrac%7Bf'%5Cleft(x%5Cright)g%5Cleft(x%5Cright)-f%5Cleft(x%5Cright)g'%5Cleft(x%5Cright)%7D%7B%5Cleft(g%5Cleft(x%5Cright)%5Cright)%5E2%7D%7B%2C%7D%5C%20kun%5C%20g%5Cleft(x%5Cright)%5Cne0" alt="D\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{\left(g\left(x\right)\right)^2}{,}\ kun\ g\left(x\right)\ne0"/></div> <div> </div> <div>Esim</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(2x%5Cleft(x%5E2%2B1%5Cright)%5Cright)%3D" alt="D\left(2x\left(x^2+1\right)\right)="/></div> <div>nyt</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%7B%2C%7D%5C%20f%5Cleft(x%5Cright)%3D2%5C%20ja%5C%20g%5Cleft(x%5Cright)%3Dx%5E2%2B1%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D2x" alt="f\left(x\right)=2x{,}\ f\left(x\right)=2\ ja\ g\left(x\right)=x^2+1{,}\ g'\left(x\right)=2x"/></div> <div>siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(2x%5Cleft(x%5E2%2B1%5Cright)%5Cright)%3D2x%5Ccdot2x%2B2%5Cleft(x%5E2%2B1%5Cright)%3D4x%5E2%2B2x%5E2%2B2%3D6x%5E2%2B2" alt="D\left(2x\left(x^2+1\right)\right)=2x\cdot2x+2\left(x^2+1\right)=4x^2+2x^2+2=6x^2+2"/></div> <div>toinen tapa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(2x%5Cleft(x%5E2%2B1%5Cright)%5Cright)%3D2x%5E3%2B2x%3D6x%5E2%2B2" alt="D\left(2x\left(x^2+1\right)\right)=2x^3+2x=6x^2+2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7Bx%5E4-2%7D%7Bx%5E4%7D" alt="D\ \frac{x^4-2}{x^4}"/></div> <div>nyt</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E4-2%3Df'%5Cleft(x%5Cright)%3D4x%5E3" alt="f\left(x\right)=x^4-2=f'\left(x\right)=4x^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3Dx%5E4%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D4x%5E3" alt="g\left(x\right)=x^4{,}\ g'\left(x\right)=4x^3"/></div> <div>siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7Bx%5E4-2%7D%7Bx%5E4%7D%3D%5Cfrac%7B4x%5E3%5Ccdot%20x%5E4-%5Cleft(x%5E4-2%5Cright)4x%5E3%7D%7B%5Cleft(x%5E4%5Cright)%5E2%7D%3D%5Cfrac%7B4x%5E7-4x%5E7%2B8x%5E3%7D%7Bx%5E8%7D%3D%5Cfrac%7B8x%5E3%7D%7Bx%5E8%7D%3D%5Cfrac%7B8%7D%7Bx%5E5%7D%7B%2C%7D%5C%20x%5Cne0" alt="D\ \frac{x^4-2}{x^4}=\frac{4x^3\cdot x^4-\left(x^4-2\right)4x^3}{\left(x^4\right)^2}=\frac{4x^7-4x^7+8x^3}{x^8}=\frac{8x^3}{x^8}=\frac{8}{x^5}{,}\ x\ne0"/></div> <div> </div> 2019-10-28T13:17:33+02:00