The Chain Rule for Functions of more than One Variable

Chain Rule for One Independent Variable

Suppose that $x=g(t)$ and $y=h(t)$ are differentiable functions of $t$ and $z=f(x,y)$ is a differentiable function of $x$ and $y$. Then $z=f(x(t),y(t))$ is a differentiable function of $t$ and

${\frac {dz}{dt}}={\frac {\partial z}{\partial x}}\cdot {\frac {dx}{dt}}+{\frac {\partial z}{\partial y}}\cdot {\frac {dy}{dt}}$

where the ordinary derivatives are evaluated at $t$ and the partial derivatives are evaluated at $(x,y)$.

Rules of taking Jacobians

If f : R^m → Rⁿ, and h(x) : R^m → R are differentiable at 'p':

• $J_{\mathbf {p} }(\mathbf {f} +\mathbf {g} )=J_{\mathbf {p} }\mathbf {f} +J_{\mathbf {p} }\mathbf {g}$
• $J_{\mathbf {p} }(h\mathbf {f} )=hJ_{\mathbf {p} }\mathbf {f} +\mathbf {f} (\mathbf {p} )J_{\mathbf {p} }h$
• $J_{\mathbf {p} }(\mathbf {f} \cdot \mathbf {g} )=\mathbf {g} ^{T}J_{\mathbf {p} }\mathbf {f} +\mathbf {f} ^{T}J_{\mathbf {p} }\mathbf {g}$

Important: make sure the order is right - matrix multiplication is not commutative!

Chain rule with Jacobian

The chain rule for functions of several variables is as follows. For f : R^m → Rⁿ and g : Rⁿ → R^p, and g o f differentiable at p, then the Jacobian is given by

$\left(J_{\mathbf {f} (\mathbf {p} )}\mathbf {g} \right)\left(J_{\mathbf {p} }\mathbf {f} \right)$

Again, we have matrix multiplication, so one must preserve this exact order. Compositions in one order may be defined, but not necessarily in the other way.

Resources

• Chain Rule, WikiBooks: Multivariable Calculus
• Multivariable Chain Rule, Harvey Mudd College
• Chain Rule With Partial Derivatives - Multivariable Calculus Video by The Organic Chemistry Tutor 2019