- #1
- 2,567
- 4
Find the (first-order) partial derivatives of the following function (where g : [itex]\mathbb{R} \to \mathbb{R}[/itex] is continuous):
[tex]f(x, y) = \int _a ^{\int _b ^y g} g.[/tex]
--------------
I got:
[tex]D_1f(x, y) = 0[/tex]
[tex]D_2f(x, y) = \frac{\partial f}{\partial y}[/tex]
[tex]= \frac{\partial }{\partial y} \int _a ^{\int _b ^y g}g[/tex]
Let G be the antiderivative of g:
[tex]= \frac{\partial }{\partial y}\left ( G \left (\int _b ^y g\right ) - G(a) \right )[/tex]
[tex]= \frac{\partial }{\partial y} G \left (\int _b ^y g\right )[/tex]
[tex]= \left ( \frac{\partial G \left (\int _b ^y g\right )}{\partial \left (\int _b ^y g\right ) }\right ) \left (\frac{\partial }{\partial y}\int _b ^y g\right )[/tex]
[tex]= g\left (\int _b ^y g\right )\frac{\partial }{\partial y}\left (G(y) - G(b) \right )[/tex]
[tex] = g\left (\int _b ^y g\right )g(y)[/tex]
[tex]f(x, y) = \int _a ^{\int _b ^y g} g.[/tex]
--------------
I got:
[tex]D_1f(x, y) = 0[/tex]
[tex]D_2f(x, y) = \frac{\partial f}{\partial y}[/tex]
[tex]= \frac{\partial }{\partial y} \int _a ^{\int _b ^y g}g[/tex]
Let G be the antiderivative of g:
[tex]= \frac{\partial }{\partial y}\left ( G \left (\int _b ^y g\right ) - G(a) \right )[/tex]
[tex]= \frac{\partial }{\partial y} G \left (\int _b ^y g\right )[/tex]
[tex]= \left ( \frac{\partial G \left (\int _b ^y g\right )}{\partial \left (\int _b ^y g\right ) }\right ) \left (\frac{\partial }{\partial y}\int _b ^y g\right )[/tex]
[tex]= g\left (\int _b ^y g\right )\frac{\partial }{\partial y}\left (G(y) - G(b) \right )[/tex]
[tex] = g\left (\int _b ^y g\right )g(y)[/tex]
Last edited: