Computing Partial Derivatives with the Chain Rule
In multivariable calculus a partial derivative tells how a function changes when one variable varies while all other variables are held constant. When the function is composed of several nested functions, the ordinary rules for differentiation no longer apply directly. The chain rule bridges that gap, allowing us to differentiate composite functions with respect to each independent variable.
1. Quick Review of Partial Derivatives
Given a function f(x, y, z), the partial derivative with respect to x is denoted f/x and is defined as
f/x = lim_{h0} [f(x+h, y, z) f(x, y, z)] / h All other variables (y, z) are treated as constants during this limit process. Similarly we define f/y and f/z.
2. The Chain Rule in One Variable
For a singlevariable composite function g(t) = f(u(t)), the derivative is
dg/dt = f'(u(t)) u'(t)
This principle generalises to many variables, but we must keep track of which variable is changing and which intermediate variables depend on it.
3. General Form of the Multivariable Chain Rule
Suppose f is a function of m intermediate variables u, u, , u, each of which in turn depends on the independent variables x, x, , x:
f = f(u(x,,x), u(x,,x), , u(x,,x))
The partial derivative of f with respect to an independent variable x is obtained by summing over every intermediate variable:
f/x = _{i=1}^{m} (f/u_i)(u_i/x) This is the same as matrix multiplication _x f = (J_u)^T _u f, where J_u is the Jacobian of the intermediate variables.
4. StepbyStep Procedure
- Identify the outer function. Write
fin terms of the intermediate variablesu_i. - Compute the partial derivatives of the outer function. Find
f/u_ifor each intermediate variable. - Differentiate each intermediate variable with respect to the independent variable. Get
u_i/x. - Multiply and sum. Form the product
(f/u_i)(u_i/x)for eachiand add them together. - Simplify. Combine like terms and factor where possible.
5. Illustrative Example
Problem: Let f(x,y) = sin( xy ). Compute f/x and f/y.
Solution:
Define the inner function u = xy. Then f = sin(u). Apply the chain rule:
f/x = cos(u)u/x,f/y = cos(u)u/y.
Now compute the inner derivatives:
u/x = 2xy,u/y = x.
Substitute back:
f/x = cos(xy)2xy,f/y = cos(xy)x.
Both partial derivatives share the factor cos(xy), illustrating the typical structure when a single inner function is involved.
6. Multiple Inner Functions
When more than one inner function appears, each contributes its own term. Consider
f(x,y) = e^{x y}ln(x+ y) Here we have two inner functions:
u = x y(used in the exponential)u = x + y(used in the logarithm)
Write f = e^{u}ln(u). Then
f/x = (/x)[e^{u}]ln(u) + e^{u}(/x)[ln(u)] Compute each piece:
e^{u}/x = e^{u}u/x = e^{x y}y,ln(u)/x = (1/u)u/x = 1/(x + y). Thus
f/x = e^{x y}yln(x + y) + e^{x y}[1/(x + y)] = e^{x y}[ yln(x + y) + 1/(x + y) ] Similarly for f/y:
u/y = x,u/y = 2yf/y = e^{x y}xln(x + y) + e^{x y}[2y/(x + y)] = e^{x y}[ xln(x + y) + 2y/(x + y) ] Notice each inner derivative appears exactly once per term, multiplied by the derivative of the outer factor that depends on it.
7. VectorValued Functions
Often we encounter a vector function r(u,v) = g(u,v), g(u,v), g(u,v). To compute the Jacobian of r with respect to (x,y) when u = u(x,y) and v = v(x,y), apply the chain rule componentwise:
r/x = g/uu/x + g/vv/x, g/uu/x + g/vv/x, g/uu/x + g/vv/x
The same pattern holds for r/y. This formulation is heavily used in physics (e.g., coordinate transformations) and in computer graphics.
8. Common Pitfalls
- Forgetting to treat other variables as constants. When computing
u_i/x, only the variablexvaries. - Missing a term. Each intermediate variable that depends on
xmust contribute a product term. Overlooking a hidden dependence leads to an incomplete derivative. - Mixing up total and partial derivatives. The chain rule presented here deals with partial derivatives; do not replace them with total derivatives unless the context explicitly requires it.
- Incorrect notation. Write
f/x(partial) rather thandf/dx(total) when more than one independent variable is present.
9. Practical Tips for HandCalculations
- Draw a dependency diagram. Nodes represent variables, arrows point from independent to dependent variables. The diagram helps visualise which
u_i/xterms are needed. - Label each arrow with the derivative it represents. The final derivative is the sum of all paths from
xtof. - When dealing with trigonometric, exponential, or logarithmic outer functions, remember their simple derivatives (
cos, sin, e^, 1/u) and multiply them by the inner derivatives. - Use a symbolic algebra system for long expressions. Most CAS tools implement the multivariable chain rule automatically.
10. Summary
The multivariable chain rule extends the familiar onedimensional rule to functions of several variables. Its core idea is simple: the rate of change of an outer function with respect to an independent variable is the sum of the rates of change of each inner variable multiplied by the rate at which that inner variable changes with respect to the independent variable.
Remember:
- Identify every intermediate variable that depends on the independent variable.
- Compute the outer partial derivative with respect to each intermediate variable.
- Multiply by the derivative of that intermediate variable with respect to the independent variable.
- Sum all contributions.
Mastering this technique unlocks the ability to differentiate complex models in physics, engineering, economics, and machine learning, where composite functions appear everywhere.
Compute Partial Derivatives With Chain Rule and Reference File Download Link
Solutions To Examples On Partial Derivatives and Reference File Download Link