Generating...                               quiz04a_n24

  1. Let $f(x,y) = \frac{y}{x-y}$. Find all the second partial derivatives.

    $f_{xx}(x,y) = \frac{2\,y}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = -\frac{x+y}{\left(-x+y\right)^3}$; $f_{yy}(x,y) = \frac{2\,x}{\left(-x+y\right)^3}$ $f_{xx}(x,y) = -\frac{2\,y}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = \frac{x+y}{\left(-x+y\right)^3}$; $f_{yy}(x,y) = -\frac{2\,x}{\left(-x+y\right)^3}$ $f_{xx}(x,y) = -\frac{2\,x}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = -\frac{y}{\left(-x+y\right)^2}$; $f_{yy}(x,y) = -\frac{2\,y}{\left(-x+y\right)^3}$ $f_{xx}(x,y) = \frac{2\,x}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = \frac{y}{\left(-x+y\right)^2}$; $f_{yy}(x,y) = \frac{2\,y}{\left(-x+y\right)^3}$

  2. Let $f(x,y) = \tan \left(x\,y^2\right)$. Find the first partial derivative $f_{ y}(x,y)$ .

    $\displaystyle 2\,x\,y\,\left(\sec \left(x\,y^2\right)\right)^2$ $\displaystyle -2\,x\,y\,\sin \left(x\,y^2\right)$ $\displaystyle -y^2\,\sin \left(x\,y^2\right)$ $\displaystyle y^2\,\left(\sec \left(x\,y^2\right)\right)^2$

  3. Let $z = f(x,y)$ where $x = t^3$ and $y = t$ . Suppose that $f_x(x,y) = y^3\,\cos \left(x\,y^3\right)$ and $f_y(x,y) = 3\,x\,y^2\,\cos \left(x\,y^3\right)$ . Find $\dfrac{dz}{dt}$.

    $\displaystyle 6\,t^5\,\cos t^6$ $\displaystyle 10\,t^9\,\cos t^{10}$ $\displaystyle -6\,t^5\,\sin t^6$ $\displaystyle -10\,t^9\,\sin t^{10}$

  4. Let $z = \frac{x\,y}{x+y}$. Find the total differential $dz$.

    $\displaystyle -\frac{-y\,dx+x\,dy}{\left(x+y\right)^2}$ $\displaystyle -\frac{dx+dy}{\left(x+y\right)^2}$ $\displaystyle \frac{y^2\,dx+x^2\,dy}{\left(x+y\right)^2}$ $\displaystyle \frac{-y\,dx+x\,dy}{\left(x+y\right)^2}$

  5. Let $f(x,y) = x^{\frac{3}{2}}\,y$. Find the tangent plane at $(x,y) = \left[ 1 , 4 \right] $.

    z = −12 + 8x + 3y z = −6 + 6x + y z = −27 + 3x + 8y z = −21 + x + 6y

  6. Let $z = f(x,y)$ where $x = r\,\sin s$ and $y = r\,\cos s$ . Express $\dfrac{\partial z}{\partial s}$ in terms of $f_x$, $f_y$, $r$ and $s$.

    $f_x ( r\,\cos s) + f_y ( -r\,\sin s)$ $f_x ( -r\,\sin s) + f_y ( r\,\cos s)$ $f_x ( \cos s) + f_y ( \sin s)$ $f_x ( \sin s) + f_y ( \cos s)$

  7. Let $z = f(x,y)$ where $x = t$ and $y = t^2$ . Suppose that $f_x(x,y) = \frac{x\,y^2\,\left(x+2\,y\right)}{\left(x+y\right)^2}$ and $f_y(x,y) = \frac{x^2\,y\,\left(2\,x+y\right)}{\left(x+y\right)^2}$ . Find $\dfrac{dz}{dt}$.

    $\displaystyle \frac{t^3\,\left(4+3\,t\right)}{\left(1+t\right)^2}$ $\displaystyle \frac{t^4\,\left(5+4\,t\right)}{\left(1+t\right)^2}$ $\displaystyle \frac{t^2\,\left(3+2\,t\right)}{\left(1+t\right)^2}$ $\displaystyle \frac{t\,\left(2+t\right)}{\left(1+t\right)^2}$

  8. Let $z = f(x,y)$ where $x = e^{s}\,\cos t$ and $y = e^{s}\,\sin t$ . Express $\dfrac{\partial z}{\partial s}$ in terms of $f_x$, $f_y$, $s$ and $t$.

    $f_x ( e^{s}\,\cos t) + f_y ( e^{s}\,\sin t)$ $f_x ( -e^{s}\,\sin t) + f_y ( e^{s}\,\cos t)$ $f_x ( e^{s}\,\cos t) + f_y ( -e^{s}\,\sin t)$ $f_x ( e^{s}\,\sin t) + f_y ( e^{s}\,\cos t)$

  9. Let $f(x,y) = -\frac{y}{x+y}$. Find the tangent plane at $(x,y) = \left[ 2 , 1 \right] $.

    $\displaystyle z=-\frac{3-x+2\,y}{9}$ $\displaystyle z=\frac{3-x+2\,y}{9}$ $\displaystyle z=\frac{-2\,x+y}{9}$ $\displaystyle z=-\frac{-2\,x+y}{9}$

  10. Let $z = x\,\sin \left(x\,y\right)$. Find the total differential $dz$.

    $\displaystyle \left(\cos \left(x\,y\right)-x\,y\,\sin \left(x\,y\right)\right)\,d x-x^2\,\sin \left(x\,y\right)\,dy$ $\displaystyle y^2\,\cos \left(x\,y\right)\,dx+\left(x\,y\,\cos \left(x\,y\right)+ \sin \left(x\,y\right)\right)\,dy$ $\displaystyle -y^2\,\sin \left(x\,y\right)\,dx+\left(\cos \left(x\,y\right)-x\,y \,\sin \left(x\,y\right)\right)\,dy$ $\displaystyle \left(x\,y\,\cos \left(x\,y\right)+\sin \left(x\,y\right)\right)\,d x+x^2\,\cos \left(x\,y\right)\,dy$


Department of Mathematics
Last modified: 2026-05-20