التعريفات والرموز للمشتقات الجزئية من الرتبة الثانية
يتم عرض أمثلة مع حلول تفصيلية حول كيفية حساب المشتقات الجزئية من الرتبة الثانية.
بالنسبة لدالة في متغيرين \( f(x, y) \)، يمكننا تعريف أربع مشتقات جزئية من الرتبة الثانية جنباً إلى جنب مع رموزها القياسية:
\[
\begin{aligned}
\dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial x} (f_x) = (f_x)_x = f_{xx} \\\\
\dfrac{\partial^2 f}{\partial y^2} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial y} (f_y) = (f_y)_y = f_{yy} \\\\
\dfrac{\partial^2 f}{\partial y \partial x} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial y} (f_x) = (f_x)_y = f_{xy} \\\\
\dfrac{\partial^2 f}{\partial x \partial y} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial x} (f_y) = (f_y)_x = f_{yx}
\end{aligned}
\]
أمثلة مع حلول تفصيلية حول المشتقات الجزئية من الرتبة الثانية
انقر على كل مثال لعرض حله التفصيلي خطوة بخطوة.
المثال 1
أوجد \( f_{xx} \) و \( f_{yy} \) بمعلومية أن \( f(x, y) = \sin(xy) \).
عرض حل المثال 1
حساب \( f_{xx} \):
\[
f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}\sin(xy)\right) = \frac{\partial}{\partial x}(y \cos(xy)) = -y^2 \sin(xy)
\]
حساب \( f_{yy} \):
\[
f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}\sin(xy)\right) = \frac{\partial}{\partial y}(x \cos(xy)) = -x^2 \sin(xy)
\]
المثال 2
أوجد \( f_{xx} \) و \( f_{yy} \) و \( f_{xy} \) و \( f_{yx} \) بمعلومية أن \( f(x, y) = x^3 + 2xy \).
عرض حل المثال 2
حساب \( f_{xx} \):
\[
f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(3x^2 + 2y) = 6x
\]
حساب \( f_{yy} \):
\[
f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(2x) = 0
\]
حساب \( f_{xy} \):
\[
f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(3x^2 + 2y) = 2
\]
حساب \( f_{yx} \):
\[
f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(2x) = 2
\]
المثال 3
أوجد \( f_{xx} \) و \( f_{yy} \) و \( f_{xy} \) و \( f_{yx} \) بمعلومية أن \( f(x, y) = x^3y^4 + x^2y \).
عرض حل المثال 3
حساب \( f_{xx} \):
\[
f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(3x^2y^4 + 2xy) = 6xy^4 + 2y
\]
حساب \( f_{yy} \):
\[
f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(4x^3y^3 + x^2) = 12x^3y^2
\]
حساب \( f_{xy} \):
\[
f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(3x^2y^4 + 2xy) = 12x^2y^3 + 2x
\]
حساب \( f_{yx} \):
\[
f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(4x^3y^3 + x^2) = 12x^2y^3 + 2x
\]