If φ = 0, φ = 0, then cos φ = 1 cos φ = 1 and ∇ f ( x 0, y 0 ) ∇ f ( x 0, y 0 ) and u u both point in the same direction. Recall that cos φ cos φ ranges from −1 −1 to 1. Therefore, the directional derivative is equal to the magnitude of the gradient evaluated at ( x 0, y 0 ) ( x 0, y 0 ) multiplied by cos φ. The ‖ u ‖ ‖ u ‖ disappears because u u is a unit vector. Therefore, the z-coordinate of the second point on the graph is given by z = f ( a + h cos θ, b + h sin θ ). The distance we travel is h h and the direction we travel is given by the unit vector u = ( cos θ ) i + ( sin θ ) j. We measure the direction using an angle θ, θ, which is measured counterclockwise in the x, y-plane, starting at zero from the positive x-axis ( Figure 4.39). Given a point ( a, b ) ( a, b ) in the domain of f, f, we choose a direction to travel from that point. We start with the graph of a surface defined by the equation z = f ( x, y ). Now we consider the possibility of a tangent line parallel to neither axis. Similarly, ∂ z / ∂ y ∂ z / ∂ y represents the slope of the tangent line parallel to the y -axis. For example, ∂ z / ∂ x ∂ z / ∂ x represents the slope of a tangent line passing through a given point on the surface defined by z = f ( x, y ), z = f ( x, y ), assuming the tangent line is parallel to the x-axis. These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change (that is, as slopes of a tangent line). A function z = f ( x, y ) z = f ( x, y ) has two partial derivatives: ∂ z / ∂ x ∂ z / ∂ x and ∂ z / ∂ y. In Partial Derivatives we introduced the partial derivative. 4.6.5 Calculate directional derivatives and gradients in three dimensions.4.6.4 Use the gradient to find the tangent to a level curve of a given function.4.6.3 Explain the significance of the gradient vector with regard to direction of change along a surface.4.6.2 Determine the gradient vector of a given real-valued function.

4.6.1 Determine the directional derivative in a given direction for a function of two variables.

Find the slope of fx equation