Multivariable Chain Rule
Derivatives through intermediate variables.
Key idea
Derivatives through intermediate variables.
In depth
If z = f(x, y) with x = x(t), y = y(t), then dz/dt = f_x dx/dt + f_y dy/dt. Add the contributions along every path from t to z. A tree diagram helps with more variables.
Worked example
z = x²y, x = t, y = t³: dz/dt = 2xy(1) + x²(3t²) = 2t⁴ + 3t⁴ = 5t⁴.
Real-world uses
- Tracking how a sensor reading changes when both time and position vary in a model
Common mistake
Forgetting one of the paths.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Multivariable Chain Rule?
Derivatives through intermediate variables. It is a topic at the expert level in Calculus.
Where is Multivariable Chain Rule used in real life?
Real-world examples: Tracking how a sensor reading changes when both time and position vary in a model.
What should I learn before Multivariable Chain Rule?
Make sure you are comfortable with Directional Derivatives, the previous topic in Calculus.
What is a common mistake with Multivariable Chain Rule?
Forgetting one of the paths.
How can I practise Multivariable Chain Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Multivariable Chain Rule?
Move on to Tangent Planes next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.