Differentiation of Implicit Functions

$x^{3}+y^{3}-3axy = 0$. Find $y'$ at $x=y$.

$x^{3}+y^{3}-3axy = 0$. Find $x$ for which $y' = 0$.

$x^{y} = y^{x}$. Find $y'$ for $x \neq y$.

$x\sin y-\cos y+\cos 2y = 0$. Find $y'$.

$x = y-\alpha \sin y$. Find $y'$ and $y''$.

$x^{2}+2xy+y^{2}-4x+2y-2 = 0$. Find $y'''$ at $x=1$, $y=1$.

$x+y = e^{x-y}$. Find $y''$.

$\ln \sqrt{x^{2}+y^{2}} = \operatorname {arctg} \frac{y}{x}$. Find $y''$.

$(x-a)^{2}+(y-b)^{2} = R^{2}$. Regarding $x$ here as the independent variable and $y$ as a function, take the fourth derivative of both sides of the equation.

For the equation $x(x^{2}+y^{2})-a(x^{2}-y^{2}) = 0$ at $x=0$, $y=0$, the formula $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}y' = 0$ does not make it possible to find $y'$; however, $y'$ can be found here too, from the formula

Find $y'$.

$(x^{2}+y^{2}-bx)^{2} = a^{2}(x^{2}+y^{2})$. Find $y'$ at $x=0$, $y=0$.

The same question for the equation $x^{3}+y^{3}-3xy = 0$.

The equations $x^{2}-y^{2}+z^{2} = 1$, $y^{2}-2x+z = 0$ are given. Find $y'$ and $z''$ at $x=1$, $y=1$, $z=1$.

Find $y'$ and $z'$.

From the equations $x^{2}+y^{2}-z^{2} = 0$, $x^{2}+2y^{2}+3z^{2} = 1$, find $d^{2}y$ and $d^{2}z$, if $x$ is the independent variable.

From the equations $x^{2}+y^{2} = 2z^{2}$, $x^{2}+2y^{2}+z^{2} = 4$, find $\frac{dx}{dz}$ and $\frac{d^{2}y}{dz^{2}}$ at the point $(1,-1,1)$, if $z$ is the independent variable.

$x+y+z = a$, $x^{3}+y^{3}+z^{3} = 3xyz$. Find the derivatives of $y$ and $z$.

At the point $(1,1,-2)$, find the first and second derivatives of $y$ and $z$, if $x+y+z = 0$, $x^{3}+y^{3}-z^{3} = 10$.

$x^{2}+y^{2}+z^{2} = 2z$. Find $\frac{\partial^{2} z}{\partial x^{2}}$.

$x^{3}+y^{3}+z^{3}-3z = 0$. Find $\frac{\partial^{2} z}{\partial x \partial y}$.

$x^{2}-2y^{2}+z^{2}-4x+2z-5 = 0$. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial^{2} z}{\partial x \partial y}$.

$x\cos y+y\cos z+z\cos x = a$. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

$xy+xz+yz = 1$. Find $dz$ and $d^{2}z$.

Find $d^{2}z$, if $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1$.

Find $d^{2}z$, if $\cos^{2} x+\cos^{2} y+\cos^{2} z = 1$.

Find $d^{2}z$ at the point $x=a$, $y=a$, $z=0$, if $x^{3}+z^{3}-3axz = y^{3}$.

Find the second partial derivatives of $z$, if this function of $x$ and $y$ is defined by the equation $y = x\varphi (z)+\psi (z)$.

Show that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z} = 0$, if $uv = 3x-2y+z$, $v^{2} = x^{2}+y^{2}+z^{2}$.

$xu+yv = 0$, $uv-xy = 5$; at $x=1$, $y=-1$ we take $u=v=2$. Find $\frac{\partial^{2} u}{\partial x^{2}}$ and $\frac{\partial^{2} u}{\partial x \partial y}$.

$x = \varphi (t)$, $y = \psi (t)$, $z = kt^{2}$. Find the derivatives of $x$ and $y$ with respect to $z$.

$x = a\cos u \sin v$, $y = b\cos u \cos v$, $z = c\sin u$. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

Show that if $F(x,y,z) = 0$, then $\frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} = -1$.

The function $u$ is given by the formula $u = x^{3}y^{2}z$, where the variables $x$, $y$, and $z$ are related by the equation $x^{2}+y^{2}+z^{2} = 3xyz$. Determine $\frac{\partial u}{\partial x}$ at the point $(1,1,1)$, when:

Show that $z$, defined as a function of $x$ and $y$ by the equation $x-az = \varphi (y-bz)$, where $\varphi$ is any differentiable function, satisfies the partial differential equation

called the equation of cylindrical surfaces.

Show that $z$, given as a function of $x$ and $y$ by the equation $z = x\varphi \left(\frac{z}{y}\right)$, satisfies the equation of conical surfaces

Show that, subject to the equations

$z$ satisfies the equation

Show that for $y = x\varphi (z)+\psi (z)$ the equation

is satisfied.

The function $z$ of $x$ and $y$ is given by the equations:

Show that

The equations are given:

Show that