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

    $\displaystyle z=-\frac{6-2\,x+y}{9}$ $\displaystyle z=\frac{6-2\,x+y}{9}$ $\displaystyle z=-\frac{-9-x+2\,y}{9}$ $\displaystyle z=\frac{-9-x+2\,y}{9}$

  2. Let $f(x,y) = \cos \left(2\,x+3\,y\right)$. Find all the second partial derivatives.

    $f_{xx}(x,y) = -9\,\sin \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = 2\,\cos \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -4\,\sin \left(2\,x+3\,y\right)$ $f_{xx}(x,y) = -4\,\sin \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = -6\,\sin \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -9\,\sin \left(2\,x+3\,y\right)$ $f_{xx}(x,y) = -4\,\cos \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = -6\,\cos \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -9\,\cos \left(2\,x+3\,y\right)$ $f_{xx}(x,y) = -9\,\cos \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = -2\,\sin \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -4\,\cos \left(2\,x+3\,y\right)$

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

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

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

    $\displaystyle \frac{t^{10}\,\left(11+9\,t^2\right)}{\left(1+t^2\right)^2}$ $\displaystyle \frac{t^4\,\left(5+3\,t^2\right)}{\left(1+t^2\right)^2}$ $\displaystyle \frac{t^8\,\left(9+7\,t^2\right)}{\left(1+t^2\right)^2}$ $\displaystyle \frac{t^2\,\left(3+t^2\right)}{\left(1+t^2\right)^2}$

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

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

  6. 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 $t$.

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

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

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

  8. Let $f(x,y) = x\,\sqrt{y}$. Find the tangent plane at $(x,y) = \left[ 4 , 1 \right] $.

    $\displaystyle z=\frac{-4+x+8\,y}{4}$ z = −2 + x + 2y $\displaystyle z=\frac{-25+8\,x+y}{4}$ z = −5 + 2x + y

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

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

  10. Let $f(x,y) = x^{y}$. Find the first partial derivative $f_{ y}(x,y)$ .

    $\displaystyle x^{-1+y}\,y$ $\displaystyle x\,e^{y}$ $\displaystyle x^{y}\,\ln x$ $\displaystyle e^{y}$


Department of Mathematics
Last modified: 2025-07-23