What are the signs of the partial derivatives?
Section 3Second-order Partial DerivativesThe partial derivative of a function of \(n\) variables, is itself a function of \(n\) variables. By taking the partial derivatives of the partial derivatives, we compute the higher-order derivatives. Higher-order derivatives are important to check the concavity of a function, to confirm whether an extreme point of a function is max or min, etc. Show Given that a function \(f(x,y)\) is continuously differentiable on an open region, we can derive the following sets of second-order partial derivatives: Direct second-order partial derivatives:\(f_{xx} = \frac{\partial f_{x}}{\partial x}\) where \(f_{x}\) is the first-order partial derivative with respect to \(x\). \(f_{yy} = \frac{\partial f_{y}}{\partial y}\) where \(f_{y}\) is the first-order partial derivative with respect to \(y\). Cross partial derivatives:\(f_{xy} = \frac{\partial f_{x}}{\partial y}\) where \(f_{x}\) is the first-order partial derivative with respect to \(x\). \(f_{yx} = \frac{\partial f_{y}}{\partial x}\) where \(f_{y}\) is the first-order partial derivative with respect to \(y\). Young's theorem: Corresponding cross partial derivatives are equal. (To read more about Young’s theorem, see Simon & Blume, Mathematics for Economists, p 330.) Suppose \(\color{red}{y = f(x_1,…,x_n)}\) is a continuously differentiable function of \(n\) variables. The first order partial derivative with respect to the variable \(\color{red}{x_{i}}\) is \(\color{red}{\partial f / \partial x_{i}}\). The \(x_{i}x_{j}\) -second order partial derivative is: \(\color{purple}{\frac{\partial }{\partial x_{j}}}(\color{red}{\frac{\partial f}{\partial x_{i}}}) = \frac{\partial^{2}f}{\partial x_{j} \partial x_{i}} = f_{i,j}\) If \(\color{red}{j = i}\), then \(x_{i}x_{j}\) -second order partial derivative is called \(\frac{\partial^{2}f}{\partial x_{i}^2}\) or second order direct partial derivatives. If \(\color{red}{j \neq i}\), then \(x_{i}x_{j}\) -second order partial derivative is called the cross partial derivatives. Example 1: Find the first, second, and cross partial derivatives for the following function: $$f(x,y) = x^{2} + 5xy + 2y^{2}$$ First order partial derivatives: $$f_{x} = 2x + 5y + 0 = 2x + 5y$$ $$f_{y} = 0 + 5x + 4y = 5x + 4y$$ Second order direct partial derivatives: $$f_{xx} = \frac{\partial }{\partial x}(2x + 5y) = 2$$ $$f_{yy} = \frac{\partial }{\partial y}(5x + 4y) = 4$$ Second-order cross partial derivatives: $$f_{xy} = \frac{\partial }{\partial y}(2x + 5y) = 5$$ $$f_{yx} = \frac{\partial }{\partial x}(5x + 4y) = 5$$ See that in this example the cross-partial derivatives are equal. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. This is not the intention of the exercise. They do not want you to indentify the graph. To explain how they want you to do it, I will do 5(a). It asks to find $f_x(1,2)$, i.e. how $f$ changes as you move in the $x$ direction from the point $(1,2)$. If you go in the positive $x$ direction from the pink spot which represents $(1,2)$, then you can see that the $z$ co-ordinate (i.e. $f$ value) increases. So the sign of $f_x$ is positive at this point. Similarly you can do 5(b), 6(a) and 6(b). When it asks to find $f_{xx}(-1,2)$, this means "the rate at which $f_x$ is changing as you advance in the $x$ direction at this point". At that point, $f_x$ is negative, since the graph is going downwards as you advance in the positive $x$ direction. But $f_x$ is increasing, since the "negative slope" is becoming less steep, like going from gradient of $-2$ to $-1$. So $f_{xx}$ is positive. Similarly you can do 7(b). When the two letters are different, you need to find "the rate at which $f_x$ changes as you move in the positive $y$ direction". This, combined with the method for question 7, is how to do 8(a) and 8(b). What is the partial derivative symbol?The symbol ∂ indicates a partial derivative, and is used when differentiating a function of two or more variables, u = u(x,t). For example means differentiate u(x,t) with respect to t, treating x as a constant.
How do you pronounce ∂?Names and coding. The symbol is variously referred to as "partial", "curly d", "rounded d", "curved d", "dabba", or "Jacobi's delta", or as "del" (but this name is also used for the "nabla" symbol ∇). It may also be pronounced simply "dee", "partial dee", "doh", or "die".
How do you know if a partial derivative is positive or negative?Second-order derivatives measure concavity, or how slope changes. Specifically: fxx: f x x : positive if the slope fx is increasing as we move in the x -direction; negative if the slope fx is decreasing as we move in the x -direction.
Is ∂ a Greek letter?Unsourced material may be challenged and removed. Delta (/ˈdɛltə/; uppercase Δ, lowercase δ or 𝛿; Greek: δέλτα, délta, [ˈðelta]) is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter dalet 𐤃.
|
