What is the formula for the directional derivative of a scalar function of two variables in the direction of a general vector $w=(a,b)$?

Conditions

• The function $f(x,y)$ is differentiable.
• The direction vector is $w=[a,b]$.
• The partial derivatives ∂$f/x$ and ∂$f/y$ exist.

Reasoning, step by step

1. Identify the components a and b of the direction vector w.
2. Calculate the partial derivative of f with respect to x.
3. Calculate the partial derivative of f with respect to y.
4. Multiply ∂$f/x$ by a and ∂f/∂y by b.
5. Sum the results to get the directional derivative.

Example

For $w=[a,b]$, the directional derivative is a*(∂f/∂x) + b*(∂$f/y$).

Common misconceptions

• Forgetting to multiply the partial derivatives by the corresponding vector components.
• Thinking the formula only works for unit vectors.

Watch the explanation

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• How is the directional derivative calculated using partial derivatives?
• What is the relationship between the directional derivative and the gradient vector?
• Why can the directional derivative be written as w·∇f?