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

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

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

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

  3. 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 $s$.

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

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

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

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

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

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

    $\displaystyle 12\,t^{11}\,\cos t^{12}$ $\displaystyle 13\,t^{12}\,\cos t^{13}$ $\displaystyle -12\,t^{11}\,\sin t^{12}$ $\displaystyle -13\,t^{12}\,\sin t^{13}$

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

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

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

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

  9. 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 $\dfrac{dz}{dt}$.

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

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

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


Department of Mathematics
Last modified: 2026-07-12