Derivative of Polar Coordinates: How to Find dy/dx

Derivative of Polar Coordinates

In polar coordinates, a point in the plane is represented by \((r, \theta)\), where:

• \(r\) is the radial distance from the origin.
• \(\theta\) is the angle from the positive \(x-axis\).

When differentiating in polar coordinates, we typically seek to find how the position of a point changes as a function of \(r\) and \(\theta\). This often requires converting between polar and Cartesian coordinates, which are related by the following equations:

\[
x = r \cos(\theta)
\]
\[y = r \sin(\theta)
\]

Derivative of Polar Coordinates with Respect to Time

If a point’s polar coordinates \((r, \theta)\) change with time, then we are interested in finding the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\). To compute these, we differentiate \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) with respect to time using the product rule and chain rule.

1. Derivative of \(x = r \cos(\theta)\):

\[
\frac{dx}{dt} = \frac{d}{dt} \left(r \cos(\theta)\right)
\]
Using the product rule:
\[
\frac{dx}{dt} = \frac{dr}{dt} \cos(\theta) – r \sin(\theta) \frac{d\theta}{dt}
\]
Thus, the rate of change of \(x\) depends on both \(\frac{dr}{dt}\) (how fast \(r\) is changing) and \(\frac{d\theta}{dt}\) (how fast \(\theta\) is changing).

2. Derivative of \(y = r \sin(\theta)\):

\[
\frac{dy}{dt} = \frac{d}{dt} \left(r \sin(\theta)\right)
\]
Using the product rule again:
\[
\frac{dy}{dt} = \frac{dr}{dt} \sin(\theta) + r \cos(\theta) \frac{d\theta}{dt}
\]

These equations show how the Cartesian components \(x\) and \(y\) change as \(r\) and \(\theta\) change with time.

Example: A Particle Moving in Polar Coordinates

Suppose a particle moves such that its radial distance changes as \(r(t) = 2t\) (i.e., \(r\) increases linearly with time), and its angular position changes as \(\theta(t) = \frac{\pi}{4}t\) (i.e., it rotates counterclockwise at a constant angular velocity). Let’s find the particle’s velocity \((\frac{dx}{dt}, \frac{dy}{dt})\) at any time \(t\).

Calculus for Beginners The Ultimate Step by Step Guide to Acing Calculus

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Step 1: Compute \(\frac{dr}{dt}\) and \(\frac{d\theta}{dt}\)

Given \(r(t) = 2t\), we differentiate with respect to time:
\[
\frac{dr}{dt} = 2
\]
Given \(\theta(t) = \frac{\pi}{4}t\), we differentiate with respect to time:
\[
\frac{d\theta}{dt} = \frac{\pi}{4}
\]

Step 2: Compute \(\frac{dx}{dt}\)

Using the formula:
\[
\frac{dx}{dt} = \frac{dr}{dt} \cos(\theta) – r \sin(\theta) \frac{d\theta}{dt}
\]
Substitute \(\frac{dr}{dt} = 2\), \(\frac{d\theta}{dt} = \frac{\pi}{4}\), and \(r(t) = 2t\):
\[
\frac{dx}{dt} = 2 \cos\left(\frac{\pi}{4}t\right) – 2t \sin\left(\frac{\pi}{4}t\right) \cdot \frac{\pi}{4}
\]
This expression represents the rate of change of the (x)-coordinate at time (t).

Step 3: Compute \(\frac{dy}{dt}\)

Using the formula:
\[
\frac{dy}{dt} = \frac{dr}{dt} \sin(\theta) + r \cos(\theta) \frac{d\theta}{dt}
\]
Substitute the known values:
\[
\frac{dy}{dt} = 2 \sin\left(\frac{\pi}{4}t\right) + 2t \cos\left(\frac{\pi}{4}t\right) \cdot \frac{\pi}{4}
\]
This expression represents the rate of change of the \(y-coordinate\) at time \(t\).

Geometrical Interpretation of the Velocity:

• Radial velocity \(\frac{dr}{dt}\) represents how fast the particle is moving toward or away from the origin (in the direction of \(r\)).
• Angular velocity \(r\frac{d\theta}{dt}\) represents how fast the particle is moving around the origin in a circular path (in the direction of \(\theta)\).

The particle’s total velocity in Cartesian coordinates can be found by combining these two components, and it shows how both the radial and angular motion contribute to the overall motion.

The Ultimate High School Math Bundle From Algebra I to Pre-Calculus

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Frequently Asked Questions

What is the polar coordinates derivative formula?

For a polar curve r = f(θ), the slope of the tangent line is dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ), where r′ = dr/dθ. It comes from writing x = r cos θ and y = r sin θ, differentiating each with the product rule, and dividing dy/dθ by dx/dθ.

How do you find polar coordinate derivatives?

Find r′, then plug r, r′ and θ into the slope formula. For r = 2 sin θ at θ = π/6, r = 1 and r′ = 2 cos(π/6) = √3. The numerator is √3(1/2) + 1(√3/2) = √3, and the denominator is √3(√3/2) − 1(1/2) = 1. So the slope of the tangent line there is √3.

Is there a derivative of polar coordinates calculator?

Yes. Graphing calculators and many free online tools can graph a polar curve and give the slope at a chosen angle. They are great for checking your work, but on a quiz or the AP Calculus BC exam you are often expected to show the steps, so practice using the formula by hand and use the calculator to confirm your answer.

What are the derivatives of the polar unit vectors?

The polar unit vectors are r̂ = ⟨cos θ, sin θ⟩ and θ̂ = ⟨−sin θ, cos θ⟩. Because they turn as θ changes, their derivatives are not zero: dr̂/dt = θ′ θ̂ and dθ̂/dt = −θ′ r̂, where θ′ = dθ/dt. These results are used in physics to find velocity and acceleration of objects moving in circles or spirals.

Are polar coordinates 3D?

No. Polar coordinates describe points in a flat, two-dimensional plane using a distance r and an angle θ. To work in three dimensions, polar coordinates are extended to cylindrical coordinates (r, θ, z), which add a height, or to spherical coordinates, which use a distance and two angles. The derivative ideas carry over in similar ways.

Recommended EffortlessMath Books

For a precalc workbook that introduces polar coordinates and parametric equations, the Pre-Calculus for Beginners covers polar conversion, polar graphs, and parametric curves with worked examples. For trig prep that polar work depends on, see Trigonometry for Beginners (resource currently unavailable).

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Area between polar curves
• Vector function derivatives
• Product rule
• Second derivatives
• Sketching curves with derivatives

Related: Absolute vs Conditional Convergence · The Rules of Integral · What are the Undefined Limits