이계 편도함수의 정의 및 표기법
이계 편도함수를 계산하는 방법에 대한 상세한 풀이가 포함된 예제들이 제시됩니다.
두 변수의 함수 \( f(x, y) \)에 대해, 표준 표기법과 함께 네 가지 이계 편도함수를 정의할 수 있습니다:
\[ \begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial x} (f_x) = (f_x)_x = f_{xx} \\\\ \dfrac{\partial^2 f}{\partial y^2} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial y} (f_y) = (f_y)_y = f_{yy} \\\\ \dfrac{\partial^2 f}{\partial y \partial x} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial y} (f_x) = (f_x)_y = f_{xy} \\\\ \dfrac{\partial^2 f}{\partial x \partial y} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial x} (f_y) = (f_y)_x = f_{yx} \end{aligned} \]이계 편도함수 상세 풀이 예제
각 예제를 클릭하여 단계별 상세 풀이를 확인하세요.
예제 1
\( f(x, y) = \sin(xy) \)일 때 \( f_{xx} \)와 \( f_{yy} \)를 구하세요.
예제 1 풀이 보기
\( f_{xx} \) 계산하기:
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}\sin(xy)\right) = \frac{\partial}{\partial x}(y \cos(xy)) = -y^2 \sin(xy) \]\( f_{yy} \) 계산하기:
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}\sin(xy)\right) = \frac{\partial}{\partial y}(x \cos(xy)) = -x^2 \sin(xy) \]예제 2
\( f(x, y) = x^3 + 2xy \)일 때 \( f_{xx} \), \( f_{yy} \), \( f_{xy} \), 및 \( f_{yx} \)를 구하세요.
예제 2 풀이 보기
\( f_{xx} \) 계산하기:
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(3x^2 + 2y) = 6x \]\( f_{yy} \) 계산하기:
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(2x) = 0 \]\( f_{xy} \) 계산하기:
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(3x^2 + 2y) = 2 \]\( f_{yx} \) 계산하기:
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(2x) = 2 \]예제 3
\( f(x, y) = x^3y^4 + x^2y \)일 때 \( f_{xx} \), \( f_{yy} \), \( f_{xy} \), 및 \( f_{yx} \)를 구하세요.
예제 3 풀이 보기
\( f_{xx} \) 계산하기:
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(3x^2y^4 + 2xy) = 6xy^4 + 2y \]\( f_{yy} \) 계산하기:
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(4x^3y^3 + x^2) = 12x^3y^2 \]\( f_{xy} \) 계산하기:
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(3x^2y^4 + 2xy) = 12x^2y^3 + 2x \]\( f_{yx} \) 계산하기:
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(4x^3y^3 + x^2) = 12x^2y^3 + 2x \]