University Calculus III — Partial Derivatives Worksheet

All 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1First-Order Partial Derivatives

   Find every first-order partial derivative of each function.

   1. $f(x,y)=3x^4{y}^2-5x{y}^3+2y$
   2. $g(x,y)=x{e}^{x{y}^2}$
   3. $h(x,y,z)=z\ln (x^2+yz)$, on the set where $x^2+yz>0$
2. Q2First-Order Partial Derivatives

   The temperature of a thin metal plate at the point $(x,y)$ is $$T(x,y)={\displaystyle \frac{120}{2+x^2+3y^2}}{}^{\circ }\text{C},$$ with $x$ and $y$ measured in centimetres.

   1. Find $T_x(1,1)$ and $T_y(1,1)$.
   2. State, with units, what each of the two numbers says about the plate.
   3. A probe at $(1,1)$ can be moved a short distance parallel to the $x$-axis or the same distance parallel to the $y$-axis. Which move changes its reading more, and by about what factor?
3. Q3Higher-Order and Mixed Partial Derivatives

   Let $f(x,y)=x^2\sin (3y)+y{e}^{2x}$. Find $f_{xx}$, $f_{xy}$, $f_{yx}$ and $f_{yy}$, computing $f_{xy}$ and $f_{yx}$ separately, and state the relation between them that your answers confirm.

4. Q4Higher-Order and Mixed Partial Derivatives

   Let $$g(x,y)=x^2{y}^3+{\displaystyle \frac{\sin \left(y^2\right)}{1+y^4}}.$$ Find $g_{yyx}$, that is ${\displaystyle \frac{{\partial }^{3}g}{\partial x\partial y\partial y}}$, with as little work as you can, and justify the order in which you chose to differentiate.

5. Q5Tangent Planes to a Graph

   Find an equation of the tangent plane to the surface $z=x^{2}y-y^3+2x$ at the point where $(x,y)=(2,1)$. Give your answer in the form $ax+by+cz=d$.

6. Q6Tangent Planes to a Graph

   Find every point on the surface $z=x^2+xy+y^2$ at which the tangent plane is parallel to the plane $6x-2z=5$, and give an equation of the tangent plane at each such point.

7. Q7Linear Approximation and Differentials

   Let $f(x,y)=\sqrt{2x+y^2}$.

   1. Find the linearization $L(x,y)$ of $f$ at $(4,1)$.
   2. Use it to approximate $f(4.06,0.97)$.
8. Q8Linear Approximation and Differentials

   The density of a solid ball of mass $m$ and radius $r$ is $\delta ={\displaystyle \frac{3m}{4\pi r^3}}$. The mass of a steel ball is measured with an error of at most $2\%$ and its radius with an error of at most $1\%$. Use differentials to estimate the maximum percentage error in the computed density, and say which of the two measurements contributes more to it.

9. Q9The Chain Rule for Several Variables

   Let $z=x^{2}y-3x{y}^2$, where $x=2t+1$ and $y=t^2$. Use the chain rule to find ${\displaystyle \frac{dz}{dt}}$ at $t=1$, without first writing $z$ as a function of $t$.

10. Q10The Chain Rule for Several Variables

    Let $z=x^{2}y+x{e}^y$, where $x=st$ and $y=s^2-4t^2$.

    1. Draw the tree diagram linking $z$ to $s$ and $t$, and write the chain rule for ${\displaystyle \frac{\partial z}{\partial s}}$ and for ${\displaystyle \frac{\partial z}{\partial t}}$.
    2. Evaluate both at $(s,t)=(2,1)$.
11. Q11Implicit Differentiation with Partial Derivatives

    The equation $x^{2}z+y{z}^3-2xy=-1$ defines $z$ implicitly as a differentiable function of $x$ and $y$ near the point $(1,2,1)$.

    1. Verify that $(1,2,1)$ satisfies the equation.
    2. Find ${\displaystyle \frac{\partial z}{\partial x}}$ and ${\displaystyle \frac{\partial z}{\partial y}}$ in terms of $x$, $y$ and $z$, and evaluate them at $(1,2,1)$.
12. Q12Synthesis — drawing on several topics in this unit

    Let $f(x,y)=y\ln x+x{y}^2$ for $x>0$.

    1. Find $f_x$ and $f_y$, and verify that $f_{xy}=f_{yx}$.
    2. Find an equation of the tangent plane to the surface $z=f(x,y)$ at the point where $(x,y)=(1,2)$.
    3. Use your answer to (b) to estimate $f(1.03,1.98)$.

The 8 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the University Calculus III bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Partial Derivatives notes cover  About the Partial Derivatives unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

A partial derivative is an ordinary derivative with the other variables frozen

Q1 asks for every first-order partial derivative of three functions, and the only new idea is the one in the heading: to find the partial with respect to x, treat y (and z) as fixed numbers and differentiate exactly as in Calculus I. Everything difficult in the question is a one-variable rule you already know — the difficulty is recognising which rule applies once the other variables have become constants.

What the number says about the object

Q2 gives the temperature of a plate as a function of position. Part (a) is Q1's technique followed by an evaluation at one point; rewriting the quotient as a constant times a negative power turns each partial into a single chain-rule step. Part (b) is the part students skip, and it is the point of the question.

Part (c) compares two short moves of the same length. For a small step, the change in a reading is about the partial derivative times the step, so with equal steps the comparison is between the sizes of the two partials, and the factor is their ratio. Compare absolute values: the question asks which move changes the reading more, not which way.

Higher-order partials, and when the order does not matter

Q3 asks for all four second-order partials of one function and insists that the two mixed ones be computed separately: one by differentiating in x and then in y, the other in the opposite order. Keep the notation straight — in the subscript form, the letters are read left to right in the order you differentiate.

Q4 turns that theorem into a strategy. It asks for a third-order partial written in an order that would have you differentiate an awkward quotient twice. The instruction "with as little work as you can" is an invitation to change the order: look at which variables each piece of the function contains, and ask which variable, differentiated first, makes the difficult piece disappear.

The tangent plane to a graph

Near a point, a smooth surface z = f(x, y) looks like a plane, and the two partial derivatives are that plane's slopes in the x- and y-directions. Q5 asks for its equation at a given point.

Q6 runs the idea backwards. Instead of a point, you are given a plane, and asked where the surface has a tangent plane parallel to it. The tool is the normal vector: rewriting the tangent plane with everything on one side shows that (f_x, f_y, −1) is normal to it.

Linear approximation: the tangent plane used as a formula

The tangent plane's equation, read as a function of x and y, is the linearization L(x, y). Q7 asks for it at a point where f is easy to evaluate, then uses it at a nearby point where it is not. This is the one-variable linearization from Calculus I with one more term.

Q8 uses the same idea to estimate an error. The density of a ball depends on its mass and its radius, each measured with a percentage error, and the question asks for the largest percentage error in the density. Write the total differential, then divide it by the density itself: for a product of powers, the relative error of the result becomes a combination of the relative errors of the inputs, each multiplied by the power that input carries.

The chain rule, one path at a time

When z depends on x and y, and x and y both depend on t, a change in t reaches z along two routes. Q9 asks for dz/dt at one value of t, and forbids the shortcut of substituting first.

Q10 has two independent variables, s and t, and asks you to draw the tree diagram before writing anything else. The tree is not decoration: z at the top, the intermediate variables x and y below it, and s and t at the bottom of every branch. A partial derivative of z with respect to s is a sum over the paths from z down to s, each path contributing the product of the derivatives along it.

Implicit differentiation, done with partial derivatives

Q11 gives an equation in x, y and z that defines z as a function of x and y near a point, with no hope of solving for z. Part (a) is a substitution check. Part (b) is the method: write the equation as F(x, y, z) = constant, differentiate it in x by the chain rule with z depending on x, and solve for ∂z/∂x.

The synthesis question

Q12 follows one function through three sheets. Part (a) is Q1 and Q3 together: two first partials, then both mixed partials computed in each order and compared. Part (b) is Q5's tangent plane at a given (x, y) — find the height first. Part (c) asks you to use part (b)'s answer, and the link is the one this set keeps returning to: the tangent plane is the graph of the linearization, so substituting the nearby point into the plane's equation is the estimate. Write the two displacements with their signs, as in Q7.

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Getting the most out of it

Circle the variable before you differentiate

Before each partial derivative, mark every factor and term that contains the variable you are differentiating in. Everything unmarked is a constant for this step. It takes seconds, and it decides whether you need the product rule, the chain rule, or neither.

Differentiate first, substitute second

Evaluating a function at a point before differentiating turns it into a number whose derivative is zero. Find each partial as a formula, then evaluate it. The same rule applies to the chain rule: build the formula, then find the intermediate values, then substitute.

Draw the tree every time

Even when a chain-rule problem looks simple enough to do in your head, draw the diagram. It shows how many terms each derivative has, and a missing path is the most common way to lose a chain-rule question.

Write units on every interpreted rate

A partial derivative of a physical quantity has the units of the output per unit of the input. Writing them down forces you to say which variable changed and which was held fixed, which is exactly what an interpretation question is marked on.

Common questions