Understanding second partial derivatives with graphs

Understanding Second Partial Derivatives with a Quadratic Function

Let’s consider a function of two variables, x and y. The function is f(x, y) = x²y + xy². This function is a combination of quadratic and linear terms.

To find the second partial derivatives, we first compute the first partial derivatives. The first partial derivative with respect to x, denoted as ∂f/∂x, is found by differentiating f(x, y) with respect to x while treating y as a constant. This gives us ∂f/∂x = 2xy + y².

Similarly, the first partial derivative with respect to y, denoted as ∂f/∂y, is found by differentiating f(x, y) with respect to y while treating x as a constant. This gives us ∂f/∂y = x² + 2xy.

Next, we compute the second partial derivatives. The second partial derivative with respect to x, denoted as ∂²f/∂x², is found by differentiating ∂f/∂x with respect to x while treating y as a constant. This gives us ∂²f/∂x² = 2y.

The second partial derivative with respect to y, denoted as ∂²f/∂y², is found by differentiating ∂f/∂y with respect to y while treating x as a constant. This gives us ∂²f/∂y² = 2x.

The mixed second partial derivatives, denoted as ∂²f/∂x∂y and ∂²f/∂y∂x, are found by differentiating ∂f/∂x with respect to y and ∂f/∂y with respect to x, respectively. In this case, they are equal, and we get ∂²f/∂x∂y = ∂²f/∂y∂x = 2x + 2y.

The second partial derivatives give us information about the curvature of the function in different directions. In this case, ∂²f/∂x² and ∂²f/∂y² tell us about the curvature in the x and y directions, respectively, while ∂²f/∂x∂y = ∂²f/∂y∂x tell us about the curvature in the direction diagonal to the x and y axes.

Exploring Second Partial Derivatives with an Exponential Function

Let’s consider a function of two variables, x and y. The function is $f(x,y)=e^{xy}$. This function is an exponential function where the exponent is a product of x and y.

To find the second partial derivatives, we first compute the first partial derivatives. The first partial derivative with respect to x, denoted as $\frac{\partial f}{\partial x}$, is found by differentiating $f(x,y)$ with respect to x while treating y as a constant. This gives us $\frac{\partial f}{\partial x}=y{e}^{xy}$.

Similarly, the first partial derivative with respect to y, denoted as $\frac{\partial f}{\partial y}$, is found by differentiating $f(x,y)$ with respect to y while treating x as a constant. This gives us $\frac{\partial f}{\partial y}=x{e}^{xy}$.

Next, we compute the second partial derivatives. The second partial derivative with respect to x, denoted as $\frac{{\partial }^{2}f}{\partial x^2}$, is found by differentiating $\frac{\partial f}{\partial x}$ with respect to x while treating y as a constant. This gives us $\frac{{\partial }^{2}f}{\partial x^2}=y^2{e}^{xy}$.

The second partial derivative with respect to y, denoted as $\frac{{\partial }^{2}f}{\partial y^2}$, is found by differentiating $\frac{\partial f}{\partial y}$ with respect to y while treating x as a constant. This gives us $\frac{{\partial }^{2}f}{\partial y^2}=x^2{e}^{xy}$.

The mixed second partial derivatives, denoted as $\frac{{\partial }^{2}f}{\partial x\partial y}$ and $\frac{{\partial }^{2}f}{\partial y\partial x}$, are found by differentiating $\frac{\partial f}{\partial x}$ with respect to y and $\frac{\partial f}{\partial y}$ with respect to x, respectively. In this case, they are equal, and we get $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}=(1+xy)e^{xy}$.

The second partial derivatives give us information about the curvature of the function in different directions. In this case, $\frac{{\partial }^{2}f}{\partial x^2}$ and $\frac{{\partial }^{2}f}{\partial y^2}$ tell us about the curvature in the x and y directions, respectively, while $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$ tell us about the curvature in the direction diagonal to the x and y axes.

The second partial derivatives give us information about the curvature of the function in different directions. If you imagine walking on the landscape of this function, ∂²f/∂x² and ∂²f/∂y² tell you about the steepness of the hills or valleys you encounter as you walk in the x and y directions, respectively. Larger values mean steeper slopes, while smaller values mean gentler slopes.

The mixed second partial derivatives, ∂²f/∂x∂y and ∂²f/∂y∂x, tell you about the steepness of the landscape as you move diagonally, that is, in both x and y directions simultaneously. If these values are positive, the landscape increases fastest when moving along the positive diagonal direction (increasing both x and y). If they are negative, the landscape increases fastest when moving along the negative diagonal direction (increasing one of x or y while decreasing the other).

In our specific case, ∂²f/∂x∂y = ∂²f/∂y∂x = (1 + xy)e^(xy) is always positive for positive x and y, indicating the landscape increases fastest along the positive diagonal direction. The exact rate of increase depends on the values of x and y due to the xy term in the expression. This means that if you are walking on this landscape, you would experience the steepest climb when moving in a direction that is a mix of both x and y.