Math Insight

Partial derivative introduction

Math 2241, Spring 2022
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Due date: March 16, 2022, 11:59 p.m.
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Total points: 1
  1. Let $f(x,y)$ be a function that depends on two variables, $x$ and $y$, defined by $$f(x,y)=3 x^{2} y^{3}.$$
    1. $f(1,2) = $

      $f(3,2) = $

      $f(a,b) = $

      $f(x,2) = $
    2. Let's define a new function $g(x)=f(x,2)=$
      .
      Calculate the derivative $\diff{g}{x} =$
      .

      Redefine $g$ to be $g(x)=f(x,5)=$
      .
      Calculate the derivative $\diff{g}{x} = $

      Redefine $g$ to be $g(x)=f(x,c)=$
      for any number $c$.
      Calculate the derivative $\diff{g}{x} = $

      Redefine $g$ to be $g(x)=f(x,y)=$
      for any number $y$.
      Calculate the derivative $\diff{g}{x} = $

    3. To calculate the partial derivative of $f$ with respect to $x$, denoted by $\displaystyle \pdiff{f}{x}$, we pretend that $y$ is a fixed number. Then, $f$ would look like the last version of function $g(x)$, since in that case, we viewed $y$ as a constant. The partial derivative of $f$ with respect to $x$ is exactly that last calculation for the derivative of $g$.

      $\displaystyle \pdiff{f}{x} = $

    4. Let's define a new function $h(y)=f(3,y)=$
      .
      Calculate the derivative $\diff{h}{y} =$
      .

      Redefine $h$ to be $h(y)=f(c,y)=$
      for any number $c$.
      Calculate the derivative $\diff{h}{y} = $

      Redefine $h$ to be $h(y)=f(x,y)=$
      for any number $x$.
      Calculate the derivative $\diff{h}{y} = $

    5. To calculate the partial derivative of $f$ with respect to $y$, denoted by $\displaystyle \pdiff{f}{y}$, we pretend that $x$ is a fixed number. Then, $f$ would look like the last version of function $h(y)$, above. The partial derivative of $f$ with respect to $y$ is exactly that last calculation for the derivative of $h$.

      $\displaystyle \pdiff{f}{y} = $

  2. Find the partial derivatives of the following polynomials.
    1. For $f(x,y)=x^{2} - y^{2}$, calculate $\pdiff{f}{x}=$
      and $\pdiff{f}{y}=$
      .
    2. For $f(x,y)=x^{2} y^{2}$, calculate $\pdiff{f}{x} =$
      and $\pdiff{f}{y} =$
      .
    3. For $g(x,y)=2 x^{2} - 3 x y + 5 y^{2}$, find
      $\pdiff{g}{x}=$
      and $\pdiff{g}{y}=$
      .

      We can evaluate the derivatives at particular values of $x$ and $y$ by plugging in numbers, just like for ordinary derivatives.

      Calculate $\pdiff{g}{x}(1,2) = $
      and $\diff{g}{y}(1,2)=$

    4. There's nothing special about the variables $x$ and $y$. We could use other variables.
      For $f(s,t)=2 s + 3 t + 5$, find $\pdiff{f}{s}=$
      and $\pdiff{f}{t}=$
      .
    5. For $g(u,v)=u v^{2} + u$, calculate $\pdiff{g}{u}=$
      and $\pdiff{g}{v}=$
      .
    6. For $h(y,z)=7 y z - 5 z^{3}$, calculate $\pdiff{h}{y}=$
      and $\pdiff{h}{z}=$
    7. Calculate $\pdiff{g}{s}$ and $\pdiff{g}{t}$ for $g(s,t) = 5 s^{3} t^{2} - 3 s^{2} t^{3}$ and evaluate at $(s,t)=(-3,1)$.
      $\pdiff{g}{s} = $
      , $\pdiff{g}{t}=$

      $\pdiff{g}{s}(-3,1) =$
      , $\pdiff{g}{t}\!(-3,1)=$

  3. If a term is constant, its derivative is zero. Similarly, if a term does not depend on a given variable, the partial derivative with respect to that variable is zero.
    1. Let $f(x) = e^x + e^2$. What is the ordinary derivative $\diff{f}{x}$?
    2. Let $f(x,y)=e^x + e^y$. What are the partial derivatives?
      $\pdiff{f}{x} =$
      , $\pdiff{f}{y}=$
    3. Let $\displaystyle f(x,y) = x^2y+ \frac{e^{y^{11}-y}-\ln(y^2+y^4)}{e^{y^{3}-1}-\ln(y+1)}$.
      Find $\pdiff{f}{x}=$

  4. We can also use the differentiation rules like the product rule and chain rule with partial derivatives. These rules work just like with ordinary derivatives. We just have to remember to treat one of the variables as though it were a constant.
    1. For $f(x,y) = \left(x + y\right) e^{x}$, calculate:
      $\pdiff{f}{x} = $

      $\pdiff{f}{y}=$
      .
    2. Let $g(s,t)= e^{s t}$. What are $\pdiff{g}{s}$ and $\pdiff{g}{t}$?
      $\pdiff{g}{s}=$

      $\pdiff{g}{t}=$
    3. If $h(u,v)=\ln{\left (2 u + 3 v \right )}$, find
      $\pdiff{h}{u}=$

      $\pdiff{h}{v}=$

      $\pdiff{h}{u}\!(1,0)=$

      $\pdiff{h}{v}\!(3,2)=$

  5. We can take partial derivatives of functions of three or more variables. When taking the partial derivative with respect to one variable, treat all other variables as constant.

    If $f(x,y,z)=x y \ln{\left (z \right )}$, find:
    $\pdiff{f}{x}=$

    $\pdiff{f}{y}=$

    $\pdiff{f}{z}=$

  6. Let $h(c,v)$ be your risk of heart disease as a function of the amount of cholesterol you eat $c$ and the amount of vegetables you eat $v$.
    1. What quantity indicates how much your risk of heart disease will increase as you increase the amount of cholesterol you eat while eating the same amount of vegetables?
    2. If (1) eating more cholesterol increases your risk of heart disease, and (2) eating more vegetables decreases this risk, describe what must be true about the partial derivative $\pdiff{h}{c}$?

      Must must be true about the partial derivative $\pdiff{h}{ v }$?
    3. Let $h(c,v)= 3 e^{5 c - 2 v}$. Calculate $\pdiff{h}{c}$ and $\pdiff{h}{v}$.
      $\pdiff{h}{c}=$

      $\pdiff{h}{v}=$