Analyze linked rates using direct derivatives or presets. Show steps, graphs, downloads, and responsive summaries. Make time based change estimates with clearer decision support.
| # | Mode | x | y | dx/dt | dy/dt | ∂z/∂x | ∂z/∂y | dz/dt |
|---|---|---|---|---|---|---|---|---|
| No calculations yet. | ||||||||
| Case | Mode | x | y | dx/dt | dy/dt | ∂z/∂x | ∂z/∂y | dz/dt |
|---|---|---|---|---|---|---|---|---|
| 1 | z = xy | 3 | 4 | 2 | -1 | 4 | 3 | 5 |
| 2 | z = x² + y² | 5 | 1 | 0.5 | 2 | 10 | 2 | 9 |
| 3 | z = x / y | 6 | 3 | 1 | 0.2 | 0.333333 | -0.666667 | 0.2 |
| 4 | z = x²y | 2 | 3 | 1.5 | 0.5 | 12 | 4 | 20 |
General chain rule: dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
Preset 1: z = x^a y^b, so ∂z/∂x = a x^(a-1) y^b and ∂z/∂y = b x^a y^(b-1)
Preset 2: z = x² + y², so ∂z/∂x = 2x and ∂z/∂y = 2y
Preset 3: z = xy, so ∂z/∂x = y and ∂z/∂y = x
Preset 4: z = x / y, so ∂z/∂x = 1 / y and ∂z/∂y = -x / y²
Preset 5: z = sin(xy), so ∂z/∂x = y cos(xy) and ∂z/∂y = x cos(xy)
Preset 6: z = e^(xy), so ∂z/∂x = y e^(xy) and ∂z/∂y = x e^(xy)
The graph assumes x(t) = x₀ + (dx/dt)t and y(t) = y₀ + (dy/dt)t over the selected time range.
dz/dt is the total rate of change of z with respect to time. It combines how z changes through x and y, and how x and y each change over time.
Use direct partials mode when you already know the evaluated values of ∂z/∂x and ∂z/∂y at the current point. Then the calculator combines them with dx/dt and dy/dt.
Preset formulas automatically compute z, ∂z/∂x, and ∂z/∂y from common multivariable expressions. This saves time and helps verify manual chain rule work.
The graph estimates how dz/dt changes over a selected time range. It assumes x and y move linearly using the entered dx/dt and dy/dt values.
Yes. You can download the calculation history as CSV and export the current result summary and history table as PDF.
Invalid results usually happen from division by zero, a zero graph step, or incompatible exponent inputs such as negative bases with fractional powers.
No. This tool focuses on fast numeric evaluation for common chain rule cases. It is designed for applied work, checks, and teaching support.
It helps model how one time based metric reacts when two changing drivers move together. That is useful for forecasting workload, pace, or sensitivity.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.