Calculus I, II and III

### Understanding the Directional Derivative The directional derivative of a function \( f \) at a point \( \mathbf{x} \) in the direction of a unit vector \( \mathbf{u} \) measures the rate at which the function \( f \) changes at \( \mathbf{x} \) as one moves in the direction \( \mathbf{u} \). Mathematically, it is defined as the dot product of the gradient of \( f \) at \( \mathbf{x} \), denoted as \( \nabla f(\mathbf{x}) \), and the unit vector \( \mathbf{u} \). ### Step-by-Step Calculation #### Step 1: Understanding the Gradient \( \nabla f(\mathbf{x}) \) The gradient \( \nabla f(\mathbf{x}) \) of a function \( f \) at a point \( \mathbf{x} \) is a vector that points in the direction of the steepest increase of \( f \). It is composed of the partial derivatives of \( f \) with respect to each component of \( \mathbf{x} \). If \( \mathbf{x} = (x_1, x_2, \ldots, x_n) \) and \( f \) is a function of \( n \) variables, then: \[ \nabla f(\mathbf{x}) = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n} \right) \] #### Step 2: Dot Product \( \nabla f(\mathbf{x}) \cdot \mathbf{u} \) The directional derivative in the direction of \( \mathbf{u} \) is given by the dot product of \( \nabla f(\mathbf{x}) \) and \( \mathbf{u} \): \[ \nabla f(\mathbf{x}) \cdot \mathbf{u} = \|\nabla f(\mathbf{x})\| \|\mathbf{u}\| \cos \theta \] Since \( \mathbf{u} \) is a unit vector, \( \|\mathbf{u}\| = 1 \). Therefore, the equation simplifies to: \[ \nabla f(\mathbf{x}) \cdot \mathbf{u} = \|\nabla f(\mathbf{x})\| \cos \theta \] #### Step 3: The Role of \( \theta \) Here, \( \theta \) is the angle between the gradient vector \( \nabla f(\mathbf{x}) \) and the direction vector \( \mathbf{u} \). The cosine of this angle, \( \cos \theta \), determines the effectiveness of the direction \( \mathbf{u} \) in capturing the rate of increase of \( f \). The value of \( \cos \theta \) ranges from -1 to 1. - \( \cos \theta = 1 \) when \( \theta = 0^\circ \), meaning \( \mathbf{u} \) is in the same direction as \( \nabla f(\mathbf{x}) \). This is the direction of the steepest ascent. - \( \cos \theta = -1 \) when \( \theta = 180^\circ \), meaning \( \mathbf{u} \) is in the exact opposite direction of \( \nabla f(\mathbf{x}) \). This is the direction of the steepest descent. - \( \cos \theta = 0 \) when \( \theta = 90^\circ \), indicating that \( \mathbf{u} \) is orthogonal to \( \nabla f(\mathbf{x}) \), and there is no change in \( f \) in this direction at \( \mathbf{x} \). #### Conclusion The directional derivative \( \nabla f(\mathbf{x}) \cdot \mathbf{u} \) quantifies the rate of change of \( f \) at \( \mathbf{x} \) in the direction \( \mathbf{u} \). The maximum rate of increase of \( f \) is achieved when \( \mathbf{u} \) aligns with \( \nabla f(\mathbf{x}) \), making \( \cos \theta = 1 \). This concept is fundamental in fields such as gradient ascent algorithms in optimization, where one seeks to maximize a function by iteratively moving in the direction of the steepest ascent.

Math

Let **u** be a unit vector. The directional derivative of *f* at **x** is given by ∇*f*(*x*) · **u** = ‖∇*f*(*x*)‖ ‖**u**‖ cos θ = ‖∇*f*(*x*)‖ cos θ where θ is the angle between the vectors ∇*f*(*x*) and **u**. The directional derivative of *f* at **x** gives the rate of increase of *f* in the direction **u**. The fastest increase is obviously obtained when cos θ = 1, i.e. when **u** has the same direction as ∇*f*(*x*).

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