Question

find the second order partial derivatives of e^x^y​

Answer 1

Let the given function be f(x,y).

$\sf f(x,y) = {e}^{ {x}^{y} } \\ \\ \longrightarrow \sf \: f(x,y) = {e}^{xy}$

$\large{\underline{\underline{\sf Partial \ Derivatives \ :}}}$

To compute partial derivatives, we differentiate with respect to one function while consideringthe other function as a constant.

Geometric Significance :

Differentiation in general is the measure of slope gradient of a function at a particular point, similarly, slope of expressions which are functions of two variables is calculated by Partial Differentiation. Suppose one of the variable as a constant and proceed with implicit differentiation.

First Order Partial Derivative w.r.t x,

$\sf \dfrac{ \partial f(x,y) }{dx} = \dfrac{\partial ( {e}^{xy} )}{ \partial x} \\ \\ \longrightarrow \sf \: \dfrac{ \partial f(x,y) }{ \partial x} = {e}^{xy} \times \dfrac{ \partial(xy)}{ \partial x} \\ \\ \longrightarrow \sf \: \dfrac{ \partial f(x,y )} { \partial x} =y {e}^{xy}$

First Order Partial Derivative w.r.t y,

$\sf \dfrac{ \partial f(x,y) }{ \partial y} = \dfrac{\partial ( {e}^{xy} )}{ \partial y} \\ \\ \longrightarrow \sf \: \dfrac{ \partial f(x,y) }{ \partial y} = {e}^{xy} \times \dfrac{ \partial(xy)}{ \partial y} \\ \\ \longrightarrow \sf \: \dfrac{ \partial f(x,y) }{ \partial y} =x {e}^{xy}$

Differentiating w.r.t x again,

$\sf \dfrac{ \partial {}^{2} \{ f(x,y) \} }{ \partial x {}^{2} } = y\dfrac{ \partial( {e}^{xy} )}{ \partial x} \\ \\ \longrightarrow \sf \: \dfrac{ \partial {}^{2} \{f(x,y) \} }{ \partial x {}^{2} } = y{e}^{xy} \times \dfrac{ \partial(xy)}{ \partial \: x} \\ \\ \longrightarrow \sf \: \dfrac{ \partial {}^{2} \{f(x,y) \} }{ \partial x {}^{2} } =y {}^{2} {e}^{xy}$

Similarly, differentiating w.r.t y again,

$\sf \dfrac{ \partial {}^{2} \{ f(x,y) \} }{ \partial y {}^{2} } = x\dfrac{ \partial( {e}^{xy} )}{ \partial y} \\ \\ \longrightarrow \sf \: \dfrac{ \partial {}^{2} \{f(x,y) \} }{ \partial y {}^{2} } = x{e}^{xy} \times \dfrac{ \partial(xy)}{ \partial \: y} \\ \\ \longrightarrow \sf \: \dfrac{ \partial {}^{2} \{f(x,y) \} }{ \partial y{}^{2} } =x {}^{2} {e}^{xy}$

Answer 2

There are three second order partial derivatives with a 2 variable function:

fxx, fyy and fxy=fyx

First, find the first order derivatives

/∂x f(x,y) = fx(x,y) = 5x4 + 2xy + 1

∂/∂y f(x,y) = fy(x,y) = x2

Now take another derivative of each, with respect to each variable

∂/∂x fx(x,y) = fxx(x,y) = 20x3 + 2y

∂/∂y fx(x,y) = fxy(x,y) = 2x

/∂x fy(x,y) = fyx(x,y) = 2x

∂/∂y fy(x,y) = fyy(x,y) = 0

$7$

Note that the mixed partial derivative is the same regardless of which order you take the derivatives in.