Q. 18 - Calculus

Question

When would you have to use the definition of the directional derivative rather than the shortcut $$D_{u}f(x_{0},y_{0})=\bigtriangledown f(x_{0},y_{0})\cdot u$$

Step-by-Step Solution

Answer

The definition of the directional derivative is used rather than the shortcut in any situation, regardless of whether $$f$$ is differentiable or not. 

Step 1. Given Information

Directional derivative shortcut, $$D_{u}f(x_{0},y_{0})=\bigtriangledown f(x_{0},y_{0})\cdot u$$ 

Step 2. Explanation

By definition, the directional derivative can be given as,

$$D_{u}f(x_{0},y_{0})=\displaystyle \lim_{h \to 0} \frac{f(x_{0}+\alpha h,y_{0}+\beta h)-f(x_{0},y_{0})}{h}$$ 

By shortcut, the directional derivative can be given as,

$$D_{u}f(x_{0},y_{0})=\bigtriangledown f(x_{0},y_{0})\cdot u$$ 

The directional derivative shortcut can only be used in cases where the function $$f$$ is differentiable.

For function $$f$$ to be differentiable, $$f$$ must be continuous at (x_{0},y_{0}) and the partial derivatives of $$f$$ must exist.

The directional derivative, by definition, has no such constraints and can be used in any situation, regardless of whether the function $$f$$ is differentiable or not.

Hence, we use the definition of the directional derivative rather than the shortcut in any condition, regardless of whether $$f$$ is differentiable or not.