University Business Mathematics — Differentiation Techniques and Elasticity Worksheet

All 11 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. Q1Derivatives of Exponential and Logarithmic Functions

   Differentiate each function, for $x>0$. Use only the derivatives of $e^x$, $b^x$, $\ln x$ and ${\log }_{b}x$ and the sum and constant-multiple rules; rewrite first where that helps.

   1. $f(x)=5e^x-8\ln x+2x^3$, and then evaluate $f^{\prime }(1)$ exactly.
   2. $g(x)=3·2^x+{\log }_{10}x$
   3. $h(x)=\ln \left({\displaystyle \frac{x^4}{e^2}}\right)$
2. Q2Derivatives of Exponential and Logarithmic Functions

   A courier company buys a delivery van for $45 000. Its book value $t$ years later is modelled by $$V(t)=45000(0.8{)}^t\text{dollars},t\ge 0.$$

   1. Find $V^{\prime }(t)$.
   2. Find $V^{\prime }(0)$ and $V^{\prime }(3)$, exactly and to the nearest dollar, and state their units. Use $\ln 0.8\approx -0.22314$.
   3. In one sentence, say what the comparison of $V^{\prime }(0)$ and $V^{\prime }(3)$ tells the owner.
3. Q3The Product and Quotient Rules

   Differentiate and simplify, for $x>0$.

   1. $f(x)=x^3{e}^x$ (factor your answer)
   2. $g(x)=(3x-2)\ln x$
   3. $h(x)={\displaystyle \frac{e^x}{x+2}}$ (factor $e^x$ out of the numerator)
   4. $k(x)={\displaystyle \frac{\ln x}{x^2}}$
4. Q4The Product and Quotient Rules

   A yoga studio sells monthly memberships. Its monthly revenue is $R=pq$, where $p$ is the membership price in dollars and $q$ the number of members, both functions of the time $t$ in months. Right now the price is $60 and is being raised by $1.50 a month, and the studio has $450$ members, a number that is falling by $6$ members a month.

   1. Is monthly revenue rising or falling right now, and at what rate?
   2. With the price still rising by $1.50 a month, how fast would membership have to be falling right now for revenue to be momentarily neither rising nor falling?
5. Q5The Chain Rule with Powers, Exponentials and Logarithms

   Differentiate each function, simplifying where a common factor allows it.

   1. $f(x)={(x^2-3x+5)}^3$
   2. $g(x)=3e^{x^2-4x}$
   3. $h(x)=\ln (x^2+2x+10)$
   4. $k(x)=\sqrt{4x+9}$, for $x>-\frac{9}{4}$
   5. $m(x)=2^{3x+1}$
6. Q6The Chain Rule with Powers, Exponentials and Logarithms

   A maker of board games finds that to sell $x$ copies a week it must set the price $$p=D(x)=80e^{-x/400}\text{dollars},x\ge 0.$$

   1. Find $D^{\prime }(400)$ exactly and to four decimal places, with units, using $e^{-1}\approx 0.3679$.
   2. Write the weekly revenue $R(x)$ and show that $R^{\prime }(x)=80e^{-x/400}(1-{\displaystyle \frac{x}{400}})$.
   3. Find $R^{\prime }(200)$ and $R^{\prime }(600)$ exactly and to the cent, using $e^{-1/2}\approx 0.60653$ and $e^{-3/2}\approx 0.22313$, and say what the sign of each tells the maker.
7. Q7Implicit Differentiation

   The weekly demand $x$ for a portable speaker and its price $p$, in dollars, satisfy $$x^2+25p^2=62500,x\ge 0,p\ge 0.$$

   1. Check that $(x,p)=(200,30)$ satisfies the relation.
   2. Differentiate implicitly with respect to $x$ to find ${\displaystyle \frac{dp}{dx}}$ in terms of $x$ and $p$.
   3. Evaluate ${\displaystyle \frac{dp}{dx}}$ at $(200,30)$ and say what it means, with units.
8. Q8Related Rates in Business Settings

   A skate shop sells $x$ pairs of skates a month at a price of $p$ dollars a pair, where $$x=1200-0.2p^2.$$ The price is $50 a pair and is being raised by $2 a month.

   1. How fast is the number of pairs sold changing?
   2. How fast is the monthly revenue $R=xp$ changing?
9. Q9Relative Rate of Change and Elasticity of Demand

   The relative rate of change of a quantity $f(t)$ is ${\displaystyle \frac{f^{\prime }(t)}{f(t)}}$; multiplied by $100$ it is the percentage rate of change.

   1. A franchise's annual revenue is $R(t)=3.2e^{0.06t}$ million dollars, $t$ years from now. Find its percentage rate of change.
   2. A café's monthly customer count is $N(t)=2000+80t$, $t$ months from now. Find its percentage rate of change at $t=5$ and at $t=20$.
   3. The café gains the same $80$ customers every month. Explain why its percentage rate of growth is nevertheless falling.
10. Q10Relative Rate of Change and Elasticity of Demand

    A climbing gym sells $x$ day passes a week at a price of $p$ dollars, where $$x=f(p)=900-p^2,0<p<30.$$ The elasticity of demand is $E(p)=-{\displaystyle \frac{p{f}^{\prime }(p)}{f(p)}}$; demand is elastic where $E(p)>1$ and inelastic where $E(p)<1$.

    1. Find $E(p)$.
    2. Find $E(10)$ and $E(20)$, and classify demand at each price.
    3. At $20, estimate the percentage change in demand if the price rises by $2\%$, and say whether weekly revenue rises or falls.
    4. Find the price at which demand has unit elasticity, exactly and to the cent, using $\sqrt{3}\approx 1.7321$.
11. Q11Synthesis — drawing on several topics in this unit

    A meal-kit company sells $x$ boxes a week at a price of $p$ dollars a box, where $$x=f(p)=\frac{3600000}{(p+20{)}^2},p>0.$$ The elasticity of demand is $E(p)=-{\displaystyle \frac{p{f}^{\prime }(p)}{f(p)}}$.

    1. Find $f^{\prime }(p)$.
    2. Show that $E(p)={\displaystyle \frac{2p}{p+20}}$.
    3. Classify demand at $10 and at $40, and say for each whether a small price increase raises or lowers weekly revenue.
    4. Find the price at which demand has unit elasticity, and the number of boxes sold there.

The 7 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

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.

Derivatives of exponential and logarithmic functions

Four new formulas carry the whole first sheet. The natural pair is the simplest: eˣ is its own derivative, and ln x differentiates to 1/x. Any other base brings in its natural logarithm, as a factor for the exponential and as a divisor for the logarithm.

Q1 restricts you to those four formulas and the sum and constant-multiple rules, which is why it says to rewrite first. Part (a) mixes an exponential, a logarithm and a power, then asks for f′(1) exactly — so keep e and ln 1 as symbols until the last line and simplify them there. Part (b) is the other-base pair. Part (c) looks like it needs a rule this sheet has not given you; it does not.

Q2 is the first formula in a business setting: a van losing value by a fixed percentage each year, written as a constant times 0.8 to the power t. The derivative is the other-base formula with the constant carried along, and ln 0.8 is given to you. Two points need care. The units of V′(t) are dollars per year — a rate, not a value — and the sign of ln 0.8 tells you before any arithmetic which way the value is moving. Part (c) asks for one sentence comparing the rate at purchase with the rate three years on: say what the comparison means for the owner, in terms of how fast the van is losing value, not just which number is larger.

The product and quotient rules

Neither a product nor a quotient differentiates factor by factor. The product rule keeps one factor and differentiates the other, twice; the quotient rule has a fixed order in its numerator, and that order is where the sign errors come from.

Q3 gives two products and two quotients, each built from a power or linear factor and an exponential or logarithm. The instructions are part of the question: (a) asks for a factored answer and (c) tells you to pull eˣ out of the numerator. Both are the same habit — the common factor is what later sets will need when they ask where a derivative is zero. In (b) and (d) the derivative of ln x is 1/x, so expect a term with an x in the denominator and clear it before you call the answer simplified.

Q4 is the product rule in its most useful business form. Revenue is price times quantity, and both are changing with time, so the rate of change of revenue has two parts: the change in price applied to the current members, and the change in members applied to the current price.

Put the four numbers the question states into that line, with the correct sign on each rate — a number that is falling has a negative rate. Part (a) asks for the direction as well as the size, and the sign of your result is the direction. Part (b) turns the same equation around: set dR/dt equal to zero, keep the price rate as it was, and solve for the membership rate. Then say it as the question asks, as a speed of falling.

The chain rule with powers, exponentials and logarithms

A composite function is differentiated from the outside in: differentiate the outer function with the inside left alone, then multiply by the derivative of the inside. The general forms below are the four this course uses.

Q5 gives one function of each kind plus a square root, and asks you to simplify where a common factor allows it. Parts (a) to (e) are a check that you can name the inside of each function in one glance: a quadratic under a cube, a quadratic in an exponent, a quadratic inside a logarithm, a linear expression under a root, a linear expression in the exponent of a base-2 power. In (d) the stated domain is exactly the set where the root is defined and its derivative exists — the derivative has the root in its denominator.

Q6 puts the chain rule into a demand function: price as a function of weekly sales, with sales divided by 400 in the exponent. Part (a) is the chain rule on eᵘ with u = −x/400; state the units as dollars of price per copy, and use the sign to say which way price moves as sales rise. Part (b) builds revenue as price times quantity — so x times the demand function — and asks you to show a given form of R′. That needs the product rule with a chain rule inside it, then eˣ-style factoring to reach the stated shape. Part (c) evaluates R′ at two sales levels; the question wants the meaning of each sign, which is whether selling one more copy a week would raise or lower revenue at that level.

Implicit differentiation

Sometimes demand is given as a relation between x and p rather than as p = f(x). Differentiating both sides with respect to x treats p as a function of x, so every term containing p picks up a factor dp/dx by the chain rule.

Q7 follows that order. Part (a) is a substitution check — a point that is not on the relation would make everything after it meaningless. Part (b) is steps 1 and 2 applied to a sum of squares. Part (c) evaluates at the given point and asks for the meaning with units: dp/dx is in dollars per speaker, and it says how the price that clears the market changes as weekly sales rise by one unit.

Related rates in business settings

A related-rates question gives one rate with respect to time and asks for another. The link is an equation between the quantities, differentiated with respect to t, so every quantity that changes with time brings its own rate.

Q8 runs the moves twice on one skate shop. Part (a) differentiates the demand equation with respect to time — the p² term gives 2p times dp/dt — and substitutes the current price and its rate. Part (b) needs the number of pairs sold right now as well as its rate, so compute x from the demand equation at the current price before you use the product rule on R = xp. The answer to (a) feeds (b); that is why the two parts are in this order.

Relative rate of change

A rate of change on its own does not say whether it is large: 80 new customers a month means something different to a small shop and a large one. The relative rate divides the rate by the size of the quantity, and times 100 it is a percentage rate of change.

Q9 compares the two kinds of growth from the first group of the course. Part (a) is an exponential model: differentiate with the chain rule, divide by the function itself, and simplify before you multiply by 100. Part (b) is a linear model, evaluated at two times. Part (c) asks you to explain the pattern in (b): look separately at what happens to the numerator f′(t) and to the denominator f(t) as t grows, and say which one is changing.

Elasticity of demand

Elasticity is a relative rate with a price on both sides: the percentage change in demand divided by the percentage change in price that caused it. With demand x = f(p), the definition the worksheet uses is the one below. The minus sign is there because demand falls as price rises, so f′(p) is negative and E(p) comes out positive.

Q10 works the definition from scratch on a quadratic demand function. Part (a) is f′(p), then the formula, then one simplification. Part (b) evaluates at two prices and classifies each by comparing the value with 1. Part (c) uses the meaning of E as a ratio of percentages: to first order, the percentage change in demand is about −E(p) times the percentage change in price, so multiply and keep the sign; then use the elasticity test above to decide the revenue question, rather than computing revenue twice. Part (d) sets E(p) = 1 and solves for p in the stated interval, checking each root against the interval before you keep it.

The synthesis question

Q11 carries the whole set on one meal-kit demand function with (p + 20)² in the denominator. Part (a) is the chain rule on a negative power — rewrite the quotient as a constant times (p + 20)⁻² first, and the quotient rule is not needed. Part (b) substitutes into the elasticity formula and asks you to show the stated form: the powers of p + 20 cancel down to a single factor. Part (c) is Q10's classification and the revenue test at two prices. Part (d) sets the given E equal to 1, solves for the price, and puts it back into the demand function for the weekly sales at that price.

Preview all 7 pages

Click any page to open the full PDF.

Getting the most out of it

Name the structure before you pick a rule

Before writing a derivative, say what the function is at its outermost level: a sum, a product, a quotient or a composite. That one word chooses the rule, and a function that fits none of them usually needs a logarithm law or an exponent law first, as Q1(c) and Q11(a) do.

Write u, v and their derivatives in a table

For every product, quotient and chain rule, list the pieces and their derivatives before assembling anything. It takes four short lines and removes almost every sign and order error the quotient rule produces.

Give every rate its units and its sign

Dollars per year, dollars per copy, members per month, pairs per month: the units say what the derivative measures, and the sign says which way the quantity is moving. Most "say what it means" parts in this set are marked on exactly those two things.

Differentiate first, substitute second

In implicit differentiation and related rates, keep the letters until the derivative is taken. Put the numbers in at the end, and check the point actually satisfies the relation.

Learn one formula for elasticity and one line for revenue

E(p) = −p·f′(p)/f(p) and R′(p) = f(p)·(1 − E(p)). With those two, every elasticity question is a derivative, a comparison with 1, and a sentence.

Common questions