Derivatives, Differentials, and the Chain Rule

This note covers derivatives and the chain rule at the level needed for multivariable calculus (MIT 18.02). It treats the differential $df$ as a linear approximation --- a machine that tells you how much $f$ changes when you make a small displacement. The deeper interpretation of $df$ as a 1-form (a linear map on vectors, living in the dual space) is covered in 1-Forms and the Dual Basis.

Derivatives: rates of change

A derivative measures how fast a function’s output changes as its input changes.

Single variable

For a function $f:\mathbb{R}\to \mathbb{R}$, the derivative at a point $x=a$ is

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$$

The output is a number --- the slope of the tangent line at $x=a$. Alternative notation: $\frac{df}{dx}{|}_{x=a}$.

Several variables: partial derivatives

For a function $f:{\mathbb{R}}^n\to \mathbb{R}$, the partial derivative with respect to $x_i$ at a point $\mathbf{a}$ is

$$\frac{\partial f}{\partial x_i}(\mathbf{a})=\underset{h\to 0}{\lim }\frac{f(\mathbf{a}+h{\mathbf{e}}_i)-f(\mathbf{a})}{h},$$

where ${\mathbf{e}}_i$ is the $i$-th standard basis vector (all zeros except a 1 in position $i$). This holds all other variables fixed and differentiates with respect to $x_i$ alone. The output is again a number.

Example. For $f(x,y)=x^{2}y+3y$:

$$\frac{\partial f}{\partial x}=2xy,\frac{\partial f}{\partial y}=x^2+3.$$

Vector-valued functions

A vector-valued function $\mathbf{r}:\mathbb{R}\to {\mathbb{R}}^n$ (such as a parametrized curve) has a derivative that is computed component by component:

$$\mathbf{r}(t)=(x(t),y(t),z(t))⟹{\mathbf{r}}^{\prime }(t)=(x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)).$$

The derivative ${\mathbf{r}}^{\prime }(t)$ is a vector --- the velocity vector of the curve at parameter $t$. See Arc-Length Parametrization for how this velocity vector relates to speed and arc length.

The differential: linear approximation

The differential $df$ captures the best linear approximation to how $f$ changes near a point.

Single variable

For $f:\mathbb{R}\to \mathbb{R}$, the differential at $x$ is

$$df=f^{\prime }(x)dx,$$

where $dx$ represents a small change in the input. The meaning: if you change $x$ by a small amount $dx$, then $f$ changes by approximately $f^{\prime }(x)dx$. This is the tangent-line approximation.

Example. If $f(x)=x^3$, then $df=3x^{2}dx$. At $x=2$, a change of $dx=0.01$ gives $df=3(4)(0.01)=0.12$. The actual change is $f(2.01)-f(2)=8.120601-8=0.120601$, so the linear approximation is very close.

Several variables

For $f:{\mathbb{R}}^n\to \mathbb{R}$, the differential is

$$df=\frac{\partial f}{\partial x_1}d{x}_1+\frac{\partial f}{\partial x_2}d{x}_2+\cdots +\frac{\partial f}{\partial x_n}d{x}_n=\sum _{i=1}^n\frac{\partial f}{\partial x_i}d{x}_i.$$

Here each $d{x}_i$ represents a small change in the $i$-th coordinate. The differential $df$ tells you: if you make small changes $d{x}_1,d{x}_2,\dots ,d{x}_n$ simultaneously, the total change in $f$ is approximately $df$.

Example. For $f(x,y)=x^{2}y+3y$:

$$df=2xydx+(x^2+3)dy.$$

At the point $(1,2)$ with displacements $dx=0.1$, $dy=-0.05$:

$$df=2(1)(2)(0.1)+(1+3)(-0.05)=0.4-0.2=\mathrm{0.2.}$$

Differential vs. derivative

A derivative ($f^{\prime }(x)$ or $\frac{\partial f}{\partial x_i}$) is a number --- a rate of change. The differential ($df$) is a linear expression in the displacements $d{x}_i$ --- it tells you how much $f$ changes for a given small displacement. The partial derivatives appear as coefficients in the differential. In the notation $df={\sum }_i\frac{\partial f}{\partial x_i}d{x}_i$, the $\frac{\partial f}{\partial x_i}$ are the coefficients and the $d{x}_i$ are the displacement variables.

A deeper view exists

At the 18.02 level, $dx$ and $dy$ are “small increments.” But they have a precise identity: they are basis elements of the dual space --- linear functions that extract components from vectors. The differential $df$ is then a linear map on vectors, not just a bookkeeping device. This deeper interpretation is covered in 1-Forms and the Dual Basis and is not needed for the chain rule or arc-length computations.

The chain rule

The chain rule tells you how to differentiate a composition of functions. It is the key tool for reparametrization (changing variables).

Single variable

If $y=f(g(t))$, then

$$\frac{dy}{dt}=f^{\prime }(g(t))\cdot g^{\prime }(t)=\frac{dy}{dg}\cdot \frac{dg}{dt}.$$

The derivative of a composition is the derivative of the outer function (evaluated at the inner function) times the derivative of the inner function.

Example. Let $y=\sin (t^2)$. Here $f(u)=\sin u$ and $g(t)=t^2$, so:

$$\frac{dy}{dt}=\cos (t^2)\cdot 2t.$$

Several variables

If $f(x_1,\dots ,x_n)$ is a function of $n$ variables, and each $x_i$ depends on a parameter $t$, then

$$\frac{df}{dt}=\frac{\partial f}{\partial x_1}\frac{d{x}_1}{dt}+\frac{\partial f}{\partial x_2}\frac{d{x}_2}{dt}+\cdots +\frac{\partial f}{\partial x_n}\frac{d{x}_n}{dt}=\sum _{i=1}^n\frac{\partial f}{\partial x_i}\frac{d{x}_i}{dt}.$$

Each term accounts for how $f$ changes because $x_i$ changes with $t$. The total rate of change is the sum of all these contributions.

Vector-valued functions

The chain rule applies component by component to vector-valued functions. If $\mathbf{r}(t)=(x(t),y(t))$ and $t=t(s)$ is a change of parameter, then

$$\frac{d\mathbf{r}}{ds}=\frac{d\mathbf{r}}{dt}\cdot \frac{dt}{ds}.$$

This is used in Arc-Length Parametrization to prove the unit-speed property: the arc-length function $s(t)$ satisfies $\frac{ds}{dt}=\Vert {\mathbf{r}}^{\prime }(t)\Vert$, so by the inverse function theorem $\frac{dt}{ds}=\frac{1}{\Vert {\mathbf{r}}^{\prime }(t)\Vert }$, and the chain rule gives

$$\frac{d\tilde{\mathbf{r}}}{ds}={\mathbf{r}}^{\prime }(t)\cdot \frac{1}{\Vert {\mathbf{r}}^{\prime }(t)\Vert }=\frac{{\mathbf{r}}^{\prime }(t)}{\Vert {\mathbf{r}}^{\prime }(t)\Vert },$$

which has magnitude 1.

The chain rule for differentials

The chain rule has a clean expression in differential notation. If $g=r^2$ where $r=\sqrt{x^2+y^2+z^2}$, then:

$$dg=2rdr.$$

But also, computing directly:

$$dg=d(x^2+y^2+z^2)=2xdx+2ydy+2zdz.$$

Setting these equal:

$$2rdr=2xdx+2ydy+2zdz,$$

which gives

$$rdr=xdx+ydy+zdz.$$

This is an equality of differentials, not of numbers. It holds for any displacement $(dx,dy,dz)$. This kind of manipulation --- applying $d$ to both sides of an equation and using linearity --- is a powerful technique for deriving relationships between differentials without going through partial derivatives explicitly.

See also

• Arc-Length Parametrization --- uses the chain rule to derive the unit-speed property
• Position Vectors and Coordinate-Free Geometry --- position vectors and affine combinations, prerequisite for parametrized curves
• Directional Derivatives and the Gradient --- directional derivatives as a special case of the chain rule
• 1-Forms and the Dual Basis --- the deeper interpretation of $dx$ and $df$ as linear maps on vectors (enrichment, beyond 18.02)