If f( x , x+y , x+y+z ) = 0 and z = z( x , y ) then  what is ∂z/∂x[math] , [/math]∂z/∂y[math] , {∂}^2z/{∂x}^2 ? [/math]
If f (x , x+y , x+y+z ) = 0 and z = z (x , y) then what is solution of these partials [math]\frac{∂z}{∂x} , \frac{∂z}{∂y} , \frac{{∂^2}z}{{∂x}^2} ?[/math]
Let [math]f(x ,y)= x^2 \arctan \left(\frac{y}{x}\right) - y^2\arctan\left(\frac{x}{y}\right).[/math] How do I show that  [math]\frac{\partial^2 f}{ \partial x \partial y}=\frac{x^2 -y^2}{x^2 + y^2}?[/math] What is the value of [math]2x^2 - 7x^2 -2x + 1[/math] if [math]x=\frac{1}{2 + \sqrt{3}}[/math]? What is the solution for [math] \lim_{n  \to \infty}((1+ \frac{1}{2n}) \times (1+ \frac{3}{2n}).... \times (1+ \frac{2n-1} {2n}))^{ \frac{1}{2n}}[/math]? How can we solve [math]x^2+\frac{3}{x}+1=x+\frac{1}{3}+\frac{4}{x}+1[/math]? If [math]\frac{1}{a} + \frac{1}{b} + \frac{1}{c} =\frac{1}{a+b+c}[/math] where [math]a+b+c \neq 0, abc \neq 0,[/math] what is the value of [math](a+b) (b+c) (c+a)?[/math] What is the value of [math]x[/math] if [math]5 ^ {x + 1} = 6 ^ y[/math] and [math]2 ^ {x + y} = 3 ^ {x - y}[/math]? If [math]y=f(x)=\sqrt{x+\sqrt{x^2+\sqrt{x+\sqrt{x^2+\sqrt{...\infty}}}}}[/math] and [math]\int f'(x)dy=g(y)+c[/math]. Then what is [math]g[/math]? How do I solve [math]\frac {d}{dx}\ln(x+\frac {1}{2x})[/math]? How can this be solved  [math] \int xsin^{-1}(x) \ dx [/math]? How do you solve for [math]x[/math] in [math]\frac{z^4y^2}3=\frac{\sqrt{px}\,b^3}{y^4}[/math]? What are the solutions of [math]x^5+5x^3+17x^2+85=0[/math]? How do I solve [math]\lim\limits_{n\to\infty}\left(\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+...+\frac{x^{2^n}}{1-x^{2^{n+1}}}\right)[/math] where [math]0<x<1[/math]? If [math] \frac{1}{1} =1, \frac{2}{2} = 1, \frac {3}{3} = 1 ... [/math] why [math] \frac{0}{0} \neq 1 [/math]? If [math]n\in \mathbb{N}[/math], what is the value of [math]\sum_{k=1}^{n}k^3\cdot\left(\frac{^{n}C_{k}}{^{n}C_{k-1}}\right) ^3 [/math]? What is the inverse function of [math]x^2-2x+1[/math] if [math]x \le 1[/math]? If [math]f(f(x)) =x^2-x+1[/math], what is [math]f(0)[/math]? How can I solve [math]\frac 1 x + \frac{2x}{x+3} = 2[/math]? What is the solution of differential equation [math]\frac{dy}{dx}= \frac{y^2+x^2+2xy+y+x} {y^2+x^2+2xy+2y+2x}?[/math] How do I solve [math]\frac{1}{4} x  + x = - 3 + \frac{1}{2} x[/math]? What are all the solutions to the equation [math] 2x^2+7x\sqrt{3} + 9 = 0?[/math] What is [math]x^2+\frac{1}{x^2}[/math], given [math]x+\frac{1}{x}= 1[/math]? How do I solve this partial differentiation equation? How do I solve [math]x+x^{1/2}=0[/math]? If [math]\tan^{2}a\tan^{2}b+\tan^{2}b\tan^{2}c+\tan^{2}a\tan^{2}c+2\tan^{2}a\times\tan^{2}b\times\tan^{2}c=1[/math] prove that [math]\sin^2a +\sin^2b+\sin^2c=1[/math]? How do I solve [math] \dfrac{5+ 2\sqrt{3}}{7+\sqrt{3}} = a- \sqrt{3b}[/math]? If [math](mn+3)^2=-|m+n+2|[/math], what is [math]m^2+n^2+(m-n)^2[/math]? What is [math]\dfrac{d}{dx}\left(\sin^{-1}(x)+2\sqrt{1-x^2}\right)[/math]? Where [math]x\in(-1,1)[/math]. If f(x) = 7 x +12, what is [math]f^{-1}[/math](x) (the inverse function)? How do I solve [math]x<\dfrac{1-2x}{x-2}[/math]? If [math]X[/math] and [math]Y[/math] are two independent discrete random variables such that [math]X\sim\text{Bin}(n,p)[/math] and [math]Y\sim\text{Bin}(m,p)[/math]. How do I interpret [math]Z=X+Y\sim\text{Bin}(m+n,p)[/math] as the sum of two functions [math]X[/math] and [math]Y[/math] ?