Related rates

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

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.

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 ⁠9/4⁠ meters per second.

This section presents an example of related rates kinematics and electromagnetic induction.

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.

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

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 the Pythagorean theorem: Express dc/dt using chain rule in terms of dx/dt and dy/dt: Substitute in x = 4 mi, y = 3 mi, dx/dt = −80 mi/hr, dy/dt = 60 mi/hr and simplify Consequently, the two vehicles are getting closer together at a rate of 28 mi/hr.

The magnetic flux through a loop of area A whose normal is at an angle θ to a magnetic field of strength B is Faraday's law of electromagnetic induction states that the induced electromotive force

is the negative rate of change of magnetic flux

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

(and therefore the electromotive force) by taking the time derivative of the flux relation If for example, the loop is rotating at a constant angular velocity ω, so that θ = ωt, then