How to Use the Chain Rule in Multivariable Calculus

The multivariable chain rule tells you how to differentiate a function whose inputs depend on other variables. The main idea is simple: follow every route by which a variable can affect the output, multiply the derivatives along that route, and add the results.

The basic formula

Suppose $z=f(x,y)$, where both $x$ and $y$ are differentiable functions of a parameter $t$, and $f$ is differentiable at the corresponding point. Then $z$ also depends on $t$, and

$$\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}.$$

The partial derivatives of $f$ measure how $z$ changes when one input changes while the other is held fixed. The derivatives $dx/dt$ and $dy/dt$ describe how the inputs change with $t$. Each product is one input’s contribution to the overall change in $z$.

If $f$ is differentiable and each $x_i=x_i(t)$ is differentiable, the same rule becomes

$$\frac{dz}{dt}=\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}\frac{dx_i}{dt}.$$

A worked example with two parameters

Let

$$z=f(x,y)=x^2+3xy, \qquad x=s+t, \qquad y=st.$$

Find $\partial z/\partial s$ and $\partial z/\partial t$. First differentiate the outer function:

$$f_x=2x+3y, \qquad f_y=3x.$$

Then differentiate the inner functions:

$$x_s=1, \quad y_s=t, \qquad x_t=1, \quad y_t=s.$$

Apply the chain rule separately for each parameter:

$$\frac{\partial z}{\partial s}=f_x x_s+f_y y_s=f_x+t f_y, \qquad \frac{\partial z}{\partial t}=f_x x_t+f_y y_t=f_x+s f_y.$$

To evaluate these at $(s,t)=(1,2)$, first find the corresponding outer inputs: $x=1+2=3$ and $y=1\cdot2=2$. Thus $f_x=2(3)+3(2)=12$ and $f_y=3(3)=9$. Therefore,

$$\left.\frac{\partial z}{\partial s}\right|_{(1,2)}=12+2(9)=30, \qquad \left.\frac{\partial z}{\partial t}\right|_{(1,2)}=12+1(9)=21.$$

You can check by substitution: $z=(s+t)^2+3(s+t)st$. Differentiating this expression directly also gives $z_s=30$ and $z_t=21$ at $(1,2)$.

The same rule in Jacobian form

For differentiable functions with several outputs, use Jacobian matrices. If $g$ maps an input vector to an intermediate vector and $f$ maps that intermediate vector to an output, then

$$J_{f\circ g}(\mathbf{u})=J_f(g(\mathbf{u}))J_g(\mathbf{u}).$$

The order matters: the Jacobian of the outer function is evaluated at the output of the inner function, and it is multiplied on the left. This is the matrix version of multiplying derivatives along a dependency path and adding contributions that reach the same output.

For example, if $g(s,t)=(x(s,t),y(s,t))$ and $f(x,y)$ is scalar-valued, then

$$J_g= \begin{bmatrix} x_s & x_t\\ y_s & y_t \end{bmatrix}, \qquad J_f=\begin{bmatrix}f_x & f_y\end{bmatrix}.$$

Their product is $J_{f\circ g}=\begin{bmatrix}z_s & z_t\end{bmatrix}$, giving the two partial-derivative formulas above.

A quick way to organize the calculation

For a longer composition, sketch a dependency diagram. For example, if $z=f(x,y)$ and $x,y$ depend on $s,t$, draw arrows from $s$ and $t$ to the intermediate variables they affect, then arrows from $x,y$ to $z$. To find $z_s$, follow only paths starting at $s$. Multiply the derivatives on each path and add the path products. This is often easier to remember than trying to memorize a large formula.

If there is just one path, such as $z=f(x)$ with $x=x(t)$, the rule reduces to the familiar single-variable form $dz/dt=f'(x)x'(t)$.

Common mistakes

• Leaving out an indirect effect: If $t$ affects both $x$ and $y$, both terms belong in $dz/dt$.
• Using ordinary derivatives for the wrong role: Use partial derivatives such as $f_x$ for the outer function, and derivatives such as $x_t$ for the inner functions.
• Substituting too early: Keep the variables symbolic until you have formed the chain-rule expression. Then substitute the corresponding intermediate values.
• Reversing Jacobian order: For $f\circ g$, the product is $J_f(g(\mathbf{u}))J_g(\mathbf{u})$, not the reverse.
• Confusing partial and total change: A partial derivative of $f$ holds its other input fixed. The derivative of a composition includes all changing inputs through the chain rule.