Q.Let $z(x,y)=x^2y+3xy^4,\ x,y\in\mathbb R$. Find the linear approximation for $z$ at $(2,-1)$.

A nonlinear function is generally hard to evaluate exactly, but near any one point its graph looks almost like a straight line -- the tangent line at that point. This is the idea behind linear approximation.

Definition (Linear Approximation). Let $f:(a,b)\to\mathbb R$ be differentiable and $x_0\in(a,b)$. The linear approximation $L$ of $f$ at $x_0$ is

This is exactly the equation of the tangent line to $y=f(x)$ at $(x_0,f(x_0))$. Because $f$ is differentiable, $f(x_0+\Delta x)\approx L(x_0+\Delta x) = f(x_0)+f'(x_0)\Delta x$ when $\Delta x$ is small, and the error $f(x)-L(x)$ shrinks to $0$ faster than $x\to x_0$ (it is $o(x-x_0)$, a consequence of the definition of the derivative as a limit).

The differential. Writing $\Delta x = dx$, the tangent-line increment is called the differential of $f$:

Geometrically, $\Delta f = f(x+dx)-f(x)$ is the actual rise along the curve, while $df=f'(x)dx$ is the rise along the tangent line; for small $dx$, $\Delta f\approx df$, but the two are generally not equal (only exactly equal when $f$ is itself linear, $f(x)=mx+c$). $df$ is a function of two independent quantities, $x$ and $dx$ -- not of $x$ alone the way the derivative is.

Differentials of standard functions follow directly from the derivative rules, e.g. $d(x^n)=nx^{n-1}dx$, $d(\sin x)=\cos x\,dx$, $d(e^x)=e^x dx$, $d(\log x)=\dfrac1x dx$. The algebraic properties of differentials mirror differentiation exactly:

• $d(c)=0$ for a constant $c$; $\ d(x)=dx$.
• $d(cf) = c\,df$.
• $d(f\pm g) = df\pm dg$.
• Product rule: $d(fg) = f\,dg + g\,df$.
• Quotient rule: $d\!\left(\dfrac fg\right) = \dfrac{g\,df-f\,dg}{g^2}$, $g\ne0$.
• Chain rule: if $h=f\circ g$, $dh = f'(g(x))\,g'(x)\,dx$.
• $d(e^{f(x)}) = e^{f(x)}f'(x)\,dx$; $\ d(\log f(x)) = \dfrac{f'(x)}{f(x)}\,dx$ (for $f(x)>0$).

Extending to several variables. For $F:A\to\mathbb R$, $A\subset\mathbb R^2$ open, and $(x_0,y_0)\in A$, the linear approximation is

and the differential is $dF = \dfrac{\partial F}{\partial x}dx + \dfrac{\partial F}{\partial y}dy$, with $dx=\Delta x,\ dy=\Delta y$. Geometrically this is the tangent plane to $z=F(x,y)$ at $(x_0,y_0)$ -- exactly as the one-variable linear approximation was a tangent line. The same pattern extends to three variables: