TI-84 gives 100 for $\frac{d}{dx}\sqrt[3]{x}\,\big|_{x=0}$

Question

My TI-84 Silver Edition is doing something strange. If $f(x)=\sqrt[3]{x}$, $\frac{d}{dx}\sqrt[3]{x}=\frac{1}{3\sqrt[3]{{x^2}}}$

At $x=0$, $\frac{d}{dx}f(0)$ is undefined. When I type $\frac{d}{dx}\sqrt[3]{x}|_{x=0}$, the calculator returns $100$. Any reason behind this?

Answer 1

You apparently are using MathPrint to type that formula. In the background the TI-84 uses the nDeriv function to calculate that.

As the "TI-84 Plus/TI-84 Plus Silver Edition" guide book (the one that goes up to page 693, not the one that goes to page 216) explains on page 60, nDeriv does its calculation using the symmetric difference quotient

$$\frac{f(x+\epsilon)-f(x-\epsilon)}{2\epsilon}$$

and if you do not specify the value of $\epsilon$ as the fourth parameter, the value 1E-3 is used.

Therefore your TI-84 is calculating

$$\frac{\sqrt[3]{0+0.001}-\sqrt[3]{0-0.001}}{2\cdot 0.001}=100$$

If you want to check this, try calculating

$$\operatorname{nDeriv}(\sqrt[3]{}(x),x,0,1E-6)$$

This is just another example showing that graphing calculators are very useful but are not to be trusted. Or, as the guidebook says at the bottom of page 60,

Because of the method used to calculate nDeriv(, the TI-84 Plus can return a false derivative value at a nondifferentiable point.