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if x^x y^y z^z=c show thatn (∂^2 z)/∂x∂y=-(1+logx)(1+logy)/(z(1+logz)^3) and hence deduce that (∂^2 z)/∂x∂y=-(xlog(ex))^-1 when x=y=z

if x^x y^y z^z=c show thatn (∂^2 z)/∂x∂y=-(1+logx)(1+logy)/(z(1+logz)^3) and hence deduce that (∂^2 z)/∂x∂y=-(xlog(ex))^-1 when x=y=z Question Solution Question if x^x y^y z^z=c show thatn (∂^2 z)/∂x∂y=-(1+logx)(1+logy)/(z(1+logz)^3) and hence deduce that (∂^2 z)/∂x∂y=-(xlog(ex))^-1 when x=y=z Type 1 Questions if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0…

if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1

if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 Question Solution Question if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 Type 1 Questions if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 Type 2 Questions if…

if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0

if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 Question Solution Question if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 Type 1 Questions if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 Type 2 Questions if…

if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu

if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu Question Solution Question if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu Type 1 Questions if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 Type 2 Questions if x^x y^y z^z=c show…

if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y)

if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) Question Solution Question if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) Type 1 Questions if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 …

if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1

if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 Problem Solution Problem if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 Type 1 Questions if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 Type 2 Questions if x^x y^y z^z=c show thatn (∂^2…

if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0

if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 Problem Solution Problem if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 Type 1 Questions if u=x^3-3xy^2+x+(e^x cosy)+1, show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))=0 if u= log((x^2+y^2)/(x+y)), show that xu_x+yu_y=1 if u=tan^(-1)(y/x), verify that (∂^2 u)/(∂y ∂x) = (∂^2 u)/(∂x ∂y) if u=e^(ax+by)f(ax-by), prove that (b ∂u/∂x)+(a ∂u/∂y)=2abu if u=1/√(x^2+y^2+z^2), then show that ((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2))=0 if u=log√(x^2+y^2+z^2), then show that (x^2+y^2+z^2)(((∂^2 u)/(∂x^2))+((∂^2 u)/(∂y^2))+((∂^2 u)/(∂z^2)))=1 Type 2 Questions if x^x y^y…

Partial Derivatives

Partial derivatives deals with the differentiation of a function of many independent variables. Partial derivatives Examples: Area of rectangle depends on its length and breadth which is defined as A(l,b) Volume of parallelepiped depends on its length, breadth and height which is defined as V(l,b,h) Where equation (1) -> partial derivative of ‘u’ with respect to ‘x’, ‘y’ being a constant variable and (2) -> partial derivative of ‘u’ with respect to ‘y’, ‘x’ being a constant variable. Type 1:…

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In-determinate Forms If f(x) at x = a assumes forms like etc., which do not represent any value are called In-determinate forms. The concept of limit gives a meaningful value for the function at overcomes these in-determinate forms. The differentiations for such forms are performed using L'Hospital’s (French Mathematician) rule. L’ Hospital’s theorem Statement If f(x) and g(x) are two functions such that Problems are categorized under three different types. Type 1 Type 2 Type 3 Note Standard limits that…

Evaluate lim(x→0) [(sin^2(π/(2-x))]^sec^2(π/(2-x))

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ArrayList is a popular data structure in Java that provides a dynamic and resizable array, unlike traditional arrays that have a fixed size. ArrayLists are widely used in programming and can be easily converted from an array. In this article, we'll look at the various ways of converting an array into an ArrayList in Java. 1. Using the Arrays.asList Method The Arrays.asList method is the simplest and most straightforward way of converting an array into an ArrayList in Java. The…

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In Java, it is often necessary to convert values from one data type to another. One common conversion is converting an integer value to a String. This can be useful when needing to display a numerical value in a text format, or when concatenating multiple values into a single string. In this blog post, we will explore the different methods available to convert an int to a String in Java. 1. Using the valueOf() Method The valueOf() method is a…

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Evaluate lim(x→0) ((a^x+b^x)/2)^(1/x) Problem Solution Problem Evaluate lim(x→0) ((a^x+b^x)/2)^(1/x) Type 1 Problems Evaluate lim (x→0) (x e^x -log(1+x)) / x^2 Evaluate lim (x→π/2) log(sin x) / (π - 2x)^2 Evaluate lim (x→0) (sinh x - x) / ( sin x - x cos x) Evaluate lim (x→π/2) log(x-π/2) / tan x Evaluate lim (x→0) [a/x - cot x/a] Evaluate lim (x→0) [1/x - log(1+x)/x^2] Type 2 Problems Evaluate lim x -> 0 (tan x - x)/(x ^ 2 * tan…

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