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How is the directional derivative calculated using partial derivatives for a specific vector v=[−1,2]v=[-1,2]?

To calculate the directional derivative for a specific vector, multiply each partial derivative by the corresponding component of the direction vector and sum them. For v=[−1,2]v=[-1,2], this means taking -1 times the partial derivative with respect to x, plus 2 times the partial derivative with respect to y.

Conditions

  • The function f(x,y)f(x,y) is differentiable.
  • The direction vector is v=[−1,2]v=[-1,2].
  • The partial derivatives ∂f/xf/x and ∂f/yf/y exist.

Reasoning, step by step

  1. Identify the components of the vector v: -1 and 2.
  2. Multiply the partial derivative ∂f/∂x by -1.
  3. Multiply the partial derivative ∂f/yf/y by 2.
  4. Add the two products together.

Example

For v=[−1,2]v=[-1,2], the directional derivative is -∂f/∂x+2x + 2∂f/∂y.

Common misconceptions

  • Mixing up which component multiplies which partial derivative.
  • Forgetting the negative sign for the x-component.

Watch the explanation

4:00 – 4:38Watch this moment ↗

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