digplanet beta 1: Athena
Share digplanet:


Applied sciences






















In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. Because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields. Because problems involve several variables, differentiation with respect to time or one of the other variables requires application of the chain rule.[1]

Fundamentally, if a function F is defined such that F=f(x), then the derivative of the function F can be taken with respect to another variable. (The Variable t is frequently used as many Related Rates problems apply to finding changes with respect to time.) We assume x is a function of t, i.e. x=g(t). Then F=f(g(t)), so

F'=f'(g(t)) \cdot g'(t)

Written in Liebnitz notation, this is:

\frac{dF}{dt} = \frac{df}{dx} \cdot \frac{dx}{dt}.

The value of this is: if it is know how x changes with respect to t, then we can determine how F changes with respect to t and vice versa. We can extend this application of the chain rule with the sum, difference, product and quotient rules of calculus, etc.


If F(x)= G(y)+ H(z)

then \frac{dF}{dx}\cdot \frac{dx}{dt}=\frac{dG}{dy} \cdot \frac{dy}{dt}+\frac{dH}{dz} \cdot \frac{dz}{dt}.


The most common way to approach related rates problems is the following:[2]

  1. Identify the known variables, including rates of change and the rate of change that is to be found. (Drawing a picture or representation of the problem can help to keep everything in order)
  2. Construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found.
  3. Differentiate both sides of the equation with respect to time (or other rate of change). Often, the chain rule is employed at this step.
  4. Substitute the known rates of change and the known quantities into the equation.
  5. Solve for the wanted rate of change.

Errors in this procedure are often caused by plugging in the known values for the variables before (rather than after) finding the derivative with respect to time. Doing so will yield an incorrect result, since if those values are substituted for the variables before differentiation, those variables will become constants; and when the equation is differentiated, zeroes appear in places of all variables for which the values were plugged in.

The "four corner" approach to solving related rates problems. Knowing the relationship between position A and position B, differentiate to find the relationship between rate A and rate B.


Leaning ladder example[edit]

A 10-meter ladder is leaning against the wall of a building, and the base of the ladder is sliding away from the building at a rate of 3 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall?

The distance between the base of the ladder and the wall, x, and the height of the ladder on the wall, y, represent the sides of a right triangle with the ladder as the hypotenuse, h. The objective is to find dy/dt, the rate of change of y with respect to time, t, when h, x and dx/dt, the rate of change of x, are known.

Step 1:


Step 2: From the Pythagorean theorem, the equation


describes the relationship between x, y and h, for a right triangle. Differentiating both sides of this equation with respect to time, t, yields


Step 3: When solved for the wanted rate of change, dy/dt, gives us


Step 4 & 5: Using the variables from step 1 gives us:


Solving for y using the Pythagorean Theorem gives:


Plugging in 8 for the equation:


It is generally assumed that negative values represent the downward direction. In doing such, the top of the ladder is sliding down the wall at a rate of 94 meters per second.

Physics Examples[edit]

Because one physical quantity often depends on another, which, in turn depends on others, such as time, related rate methods have broad applications in Physics. This section presents an example of related rates kinematics and electromagnetic induction.

Physics Example I: Relative Kinematics of Two Vehicles[edit]

One vehicle is headed North and currently located at (0,3); the other vehicle is headed West and currently located at (4,0). The chain rule can be used to find whether they are getting closer or further apart.

For example, one can consider the kinematics problem where one vehicle is heading West toward an intersection at 80 miles per hour while another is heading North away from the intersection at 60 miles per hour. One can ask whether the vehicles are getting closer or further apart and at what rate at the moment when the North bound vehicle is 3 miles North of the intersection and the West bound vehicle is 4 miles East of the intersection.

Big idea: use chain rule to compute rate of change of distance between two vehicles.


  1. Choose coordinate system
  2. Identify variables
  3. Draw picture
  4. Big idea: use chain rule to compute rate of change of distance between two vehicles
  5. Express c in terms of x and y via Pythagorean theorem
  6. Express dc/dt using chain rule in terms of dx/dt and dy/dt
  7. Substitute in x, y, dx/dt, dy/dt
  8. Simplify.

Choose coordinate system: Let the y-axis point North and the x-axis point East.

Identify variables: Define y(t) to be the distance of the vehicle heading North from the origin and x(t) to be the distance of the vehicle heading West from the origin.

Express c in terms of x and y via Pythagorean theorem:

c = (x^2 + y^2)^{1/2}

Express dc/dt using chain rule in terms of dx/dt and dy/dt:

\frac{dc}{dt} = \frac{d}{dt}(x^2 + y^2)^{1/2} Apply derivative operator to entire function
= \frac{1}{2}(x^2 + y^2)^{-1/2}\frac{d}{dt}(x^2 + y^2) Square root is outside function; Sum of squares is inside function
=\frac{1}{2}(x^2 + y^2)^{-1/2}\left[\frac{d}{dt}(x^2) + \frac{d}{dt}(y^2) \right] Distribute differentiation operator
= \frac{1}{2}(x^2 + y^2)^{-1/2}\left[ 2x\frac{dx}{dt} + 2y\frac{dy}{dt}\right] Apply chain rule to x(t) and y(t)}
= \frac{x\frac{dx}{dt} + y\frac{dy}{dt}}{\sqrt{x^2 + y^2}} Simplify.

Substitute in x = 4 mi, y = 3 mi, dx/dt = -80 mi/hr, dy/dt = 60 mi/hr and Simplify

\frac{dc}{dt} & = \frac{4 mi \cdot (-80 mi/hr) + 3 mi \cdot (60) mi/hr}{\sqrt{(4 mi)^2 + (3 mi)^2}}\\
& = \frac{-320 mi^2/hr + 180 mi^2/hr}{5 mi}\\
&= \frac{-140 mi^2/hr}{5 mi}\\
& = -28 mi/hr\\

Consequently, the two vehicles are getting closer together at a rate of 28 mi/hr.

Physics Example II: Electromagnetic induction of conducting loop spinning in magnetic field[edit]

The magnetic flux through a loop of area A whose normal is at an angle θ to a magnetic field of strength B is

 \Phi_B = B A \cos(\theta),

Faraday's law of electromagnetic induction states that the induced electromotive force \mathcal{E} is the negative rate of change of magnetic flux \Phi_B through a conducting loop.

 \mathcal{E} = -{{d\Phi_B} \over dt},

If the loop area A and magnetic field B are held constant, but the loop is rotated so that the angle θ is a known function of time, the rate of change of θ can be related to the rate of change of \Phi_B (and therefore the electromotive force) by taking the time derivative of the flux relation

\mathcal{E} = -\frac{d\Phi_B}{dt} =  B A \sin{\theta} \frac{d\theta}{dt}

If for example, the loop is rotating at a constant angular velocity ω, so that θ=ωt, then

\mathcal{E}= \omega B A \sin{\omega t}


  1. ^ "Related Rates". Whitman College. Retrieved 2013-10-27. 
  2. ^ Kreider, Donald. "Related Rates". Dartmouth. Retrieved 2013-10-27. 

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Related_rates — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
1000000 videos foundNext > 

Related rates of water pouring into cone

More free lessons at: http://www.khanacademy.org/video?v=Xe6YlrCgkIo.

Related Rates #1 Problem Using Implicit Differentiation

Buy my book!: '1001 Calculus Problems for Dummies' - you can get it on my website: http://patrickjmt.com/ Need a LIVE tutor to help answer a question?

Calculus 1 Lecture 2.8: Related Rates

Calculus 1 Lecture 2.8: Related Rates.

Related Rates

To get more help with calculus, check out: https://www.calculusexpert.com/ Understand one of the trickiest applications of derivatives and implicit differentiation, ...

Related Rates # 7 - Ladder Sliding Down Wall, Finding Rate of Change of Area Under Ladder

Related Rates # 7 - Ladder Sliding Down Wall, Finding Rate of Change of Area Under Ladder. In this example, a ladder is being pulled away from a wall ...

Related Rates in Calculus Part 1

I work through multiple examples of Related Rate problems from Calculus. In the first example I keep saying speed instead of velocity and I say that speed can ...

Falling ladder related rates

More free lessons at: http://www.khanacademy.org/video?v=kBVDSu7v8os.

Calculus I - Related Rates - Example 3 - Street Light and Shadow

Related rates problem about a man's shadow as he walks away from a street light. The related rates worksheet with the general process and examples 1 - 6 can ...

Related Rates: What you must NOT forget -- Calculus -- ThatTutorGuy.com

Calculus got you down? For Chris's videos covering the rest of calculus check out http://www.ThatTutorGuy.com.

Related Rates #2 Using Cones

Related Rates using Cones. Gravel is being dumped into a pile that forms a cone; how quickly is the height changing? A problem from UT-Austin calculus 1 ...

1000000 videos foundNext > 

154 news items

Jerusalem Post Israel News

Jerusalem Post Israel News
Sat, 28 Nov 2015 11:10:11 -0800

Today's topic is related rates, which is the most difficult topic in calculus. In fact, many students fail calculus just because they don't understand related rates.” It seemed that my response had made the student angry. “You're going to fail me just ...

Chicago Tribune

Chicago Tribune
Thu, 19 Nov 2015 16:48:08 -0800

The city's utility fund also has a surplus, and Councilwoman Tish Powell suggested tapping into those instead of increasing related rates. Stegall said that he would be developing a scenario to that end, in part because of an upcoming renewal of a deal ...

UC Food Observer

UC Food Observer
Wed, 11 Nov 2015 15:40:09 -0800

“I think what people in my generation can do is become actively engaged in the conversation. They can engage with their members of Congress and work with organizations that want to promote the health of the whole family. That's the only way we're going ...
Financial News (subscription)
Mon, 02 Nov 2015 16:16:51 -0800

... work on which started with the closure of its proprietary trading desks 10 years ago. The steps being taken now, Reuther said, involve moving all flow-related rates and FX activities to Frankfurt, leaving the bank's London team to focus on ...


Fri, 06 Apr 2012 10:10:17 -0700

Watch more at http://www.educator.com/mathematics/c... Other subjects include Biology, Chemistry, Physics, Organic Chemistry, Computer Science, Statistics, Algebra 1/2, Basic Math, Pre Calculus, Geometry, and Pre Algebra. -All lectures are broken down ...

Star Local Media

Star Local Media
Sat, 19 Sep 2015 16:38:52 -0700

Water costs and related rates likely will keep rising, whether lake levels are or not, and North Texas must brace for it. “Our program is one of the model programs in the state,” Kula said. “It's regional service through unity – we're all in this ...

PolicyShop (blog)

PolicyShop (blog)
Fri, 21 Aug 2015 08:34:20 -0700

Don't forget in the meantime we have a real unemployment rate that's probably 21%. It's not 6. [It's] not 5.2 and 5.5. Our real unemployment rate—in fact, I saw a chart the other day, our real unemployment—because you have ninety million people that ...
Mondaq News Alerts (registration)
Fri, 30 Oct 2015 07:42:31 -0700

LIBOR cannot of course be viewed in isolation, and beyond the investigation into Euribor, Swiss Franc LIBOR and other related rates, there has been investigation into the alleged fixing of other benchmark rates including ISDA and, most prominently, Forex.

Oops, we seem to be having trouble contacting Twitter

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!

Searchlight Group

Digplanet also receives support from Searchlight Group. Visit Searchlight