Differentiation in several variables · Definition

Differentiability in several variables

multivariable calculusderivatives

Before you start you need to know

It is the condition that makes «linear approximation» honest: without it, the plane built from the partials may exist and still approximate nothing.

Definition

A function \(f(x,y)\) is differentiable at a point \((a,b)\) if its graph has a tangent plane there: the error of the linear approximation is negligible compared with the distance to the point.

\[\lim_{(h,k)\to(0,0)}\frac{f(a+h,b+k)-f(a,b)-f_x(a,b)\,h-f_y(a,b)\,k}{\sqrt{h^2+k^2}}=0\]

The definition asks that the error \(f(a+h,b+k)-L(h,k)\), where \(L\) is the linear part built from the partials, be small even compared with the distance \(\sqrt{h^2+k^2}\). In one variable this is equivalent to the derivative existing; in two variables, having partials is weaker, since it only measures what happens along the axes. For example, \(f(x,y)=\frac{x^2y}{x^2+y^2}\) (with \(f(0,0)=0\)) has \(f_x(0,0)=f_y(0,0)=0\), but along \(h=k\) the error quotient equals \(\sqrt2/4\), not zero.

In practice the limit is not checked: one uses the sufficient criterion that the partials exist and are continuous on a neighborhood of the point. Hence every polynomial, exponential, sine and quotient with nonvanishing denominator is differentiable where defined. The troublesome points are vertices and creases, such as the origin for \(\sqrt{x^2+y^2}\) or \(|x|+|y|\).

Differentiability implies continuity, and it also justifies the differential \(df=f_x\,dx+f_y\,dy\): for small increments, \(\Delta f\approx df\). With \(f=x^2y\) at \((1,2)\) and \((dx,dy)=(0.1,-0.1)\) one gets \(df=4(0.1)+1(-0.1)=0.3\), close to the actual change \(0.299\).

It is the hypothesis behind what follows: the gradient, the directional derivative and the chain rule are stated for differentiable functions.

Think about it

Why does the existence of \(f_x\) and \(f_y\) at a point not guarantee that \(f\) is continuous there?

The partials only look at the two lines parallel to the axes. A function can be zero along both axes and nonzero on the diagonals, so it can be discontinuous at the point even though both partials exist.

Exercises 10

  1. Let \(f(x,y)=x^2y\). Use the tangent plane at \((1,2)\) to approximate \(f(1.1,\,1.9)\).
  2. For \(f(x,y)=x^2+3xy\) at \((1,1)\), compute the differential \(df=f_x\,dx+f_y\,dy\) with \(dx=0.1\), \(dy=-0.2\).
  3. Let \(f(x,y)=xe^{y}\). At \((2,0)\), with \(dx=0.05\) and \(dy=0.1\), what is \(df\)?
  4. Find the linearization of \(f(x,y)=\sin(xy)\) at the point \((1,0)\).
  5. Which condition guarantees that \(f\) is differentiable at \((a,b)\)?
  6. If \(f\) is differentiable at \(P\), which statement is always true?
  7. Let \(f(x,y)=\sqrt{x^2+y^2}\). Why is it not differentiable at the origin?
  8. Which of these functions is differentiable on all of \(\mathbb{R}^2\)?
  9. Let \(f(x,y)=\dfrac{x^2y}{x^2+y^2}\) with \(f(0,0)=0\). Which statement is correct?
  10. For the same \(f\) (with \(f_x(0,0)=f_y(0,0)=0\)), what value does the error quotient \(\dfrac{f(h,k)}{\sqrt{h^2+k^2}}\) approach along the line \(h=k>0\)?

References

  • James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021. §14.4
  • Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023. §14.6
  • Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012. §2.3