13.6-bo'lim: Zanjir qoidasi

1. Quyidagi ma'lumotlarni hisobga olgan holda \(\displaystyle \frac{{dz}}{{dt}}\) ni aniqlash uchun zanjir qoidasidan foydalaning. \[z = \cos \left( {y\,{x^2}} \right)\,\hspace{0.5in}x = {t^4} - 2t,\,\,\,\,y = 1 - {t^6}\] Yechim
2. Quyidagi ma'lumotlarni hisobga olgan holda \(\displaystyle \frac{{dw}}{{dt}}\) ni aniqlash uchun zanjir qoidasidan foydalaning. \[w = \frac{{{x^2} - z}}{{{y^4}}}\,\hspace{0.5in}x = {t^3} + 7,\,\,\,\,y = \cos \left( {2t} \right),\,\,\,\,z = 4t\] Yechim
3. Quyidagi ma'lumotlarni hisobga olgan holda \(\displaystyle \frac{{dz}}{{dx}}\) ni aniqlash uchun zanjir qoidasidan foydalaning. \[z = {x^2}{y^4} - 2y\,\hspace{0.5in}y = \sin \left( {{x^2}} \right)\] Yechim
4. Quyidagi ma'lumotlarni hisobga olgan holda \(\displaystyle \frac{{\partial z}}{{\partial u}}\) va \(\displaystyle \frac{{\partial z}}{{\partial v}}\) ni aniqlash uchun zanjir qoidasidan foydalaning. \[z = {x^{ - 2}}{y^6} - 4x\,\hspace{0.5in}x = {u^2}v,\,\,\,\,y = v - 3u\] Yechim
5. Quyidagi ma'lumotlarni hisobga olgan holda, \({z_t}\) va \({z_p}\) ni aniqlash uchun zanjir qoidasidan foydalaning. \[z = 4y\sin \left( {2x} \right)\,\hspace{0.5in}x = 3u - p,\,\,\,\,y = {p^2}u,\,\,\,\,\,\,u = {t^2} + 1\] Yechim
6. Quyidagi ma'lumotlarni hisobga olgan holda, \(\displaystyle \frac{{\partial w}}{{\partial t}}\) va \(\displaystyle \frac{{\partial w}}{{\partial s}}\) ni aniqlash uchun zanjir qoidasidan foydalaning. \[w = \sqrt {{x^2} + {y^2}} + \frac{{6z}}{y}\,\hspace{0.5in}x = \sin \left( p \right),\,\,\,\,y = p + 3t - 4s,\,\,\,\,z = \frac{{{t^3}}}{{{s^2}}},\,\,\,\,p = 1 - 2t\] Yechim
7. Quyidagi vaziyat uchun \(\displaystyle \frac{{\partial w}}{{\partial t}}\) va \(\displaystyle \frac{{\partial w}}{{\partial v}}\) formulalarini aniqlang. \[w = w\left( {x,y} \right)\hspace{0.5in}x = x\left( {p,q,s} \right),\,\,\,\,y = y\left( {p,u,v} \right),\,\,\,\,s = s\left( {u,v} \right),\,\,\,\,p = p\left( t \right)\] Yechim
8. Quyidagi vaziyat uchun \(\displaystyle \frac{{\partial w}}{{\partial t}}\) va \(\displaystyle \frac{{\partial w}}{{\partial u}}\) formulalarini aniqlang. \[w = w\left( {x,y,z} \right)\hspace{0.5in}x = x\left( t \right),\,\,\,\,y = y\left( {u,v,p} \right),\,\,\,\,z = z\left( {v,p} \right),\,\,\,\,v = v\left( {r,u} \right),\,\,\,\,p = p\left( {t,u} \right)\] Yechim
9. Quyidagi tenglama uchun \(\displaystyle \frac{{dy}}{{dx}}\) ni hisoblang. \[{x^2}{y^4} - 3 = \sin \left( {xy} \right)\] Yechim
10. Quyidagi tenglama uchun \(\displaystyle \frac{{\partial z}}{{\partial x}}\) va \(\displaystyle \frac{{\partial z}}{{\partial y}}\) ni hisoblang. \[{{\bf{e}}^{z\,y}} + x{z^2} = 6x{y^4}{z^3}\] Yechim
11. Quyidagi vaziyat uchun \({f_{u\,u}}\) ni aniqlang. \[f = f\left( {x,y} \right)\hspace{0.5in}x = {u^2} + 3v,\,\,\,\,\,\,\,y = uv\] Yechim