Concept: *The Magnitude of the Derivative

Prerequisite concepts

Differential Calculus Difference / Change

Understanding both difference (how far apart two values are at one time) and change (how far apart the value of a single parameter is at two different times) is necessary for understanding derivatives.

*The Magnitude of the Derivative

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Representations used

lower anchor MTH 251

The magnitude of the derivative at a point is the slope of a tangent line at that point

1D Graph

The value of the derivative at a point on a curve is equivalent to the slope of a line tangent to the curve at that point. The tangent line can be thought of as the limit of a secant line between two points infinitesimally close together

Difference / Change The derivative relates how much $f$ changes as $x$ changes The derivative is a ratio of small changes The derivative can be approximated by the slope of a secant line The derivative is a limit The derivative is a function The derivative of a constant is zero The derivative is a linear operator Power law The derivative at a cusp is undefined Variables can be held constant Product rule Single variable chain rule You can flip a derivative "With respect to what" matters

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The magnitude of the derivative is related to the density of contour lines

Contour Maps

The magnitude of the derivative of a function is directly related to how close together the contour lines of the function lie: the closer together the contour lines, the greater the magnitude of the derivative.

The Master Formula

*The Magnitude of the Gradient

The magnitude of the gradient is the value of the slope in the direction of greatest increase

The gradient is a vector *The Direction of the Gradient

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*The Magnitude of a Partial Derivative

FIXME: Add for Contours: "The magnitude of a partial derivative is proportional to the density of contour lines along the path of the partial derivative."[]FIXME: Add for Surfaces: "The magnitude of a partial derivative is the slope of the surface along the path of the partial derivative."[]FIXME: Add current concept: "All partial derivatives can be found as a slope of a tangent plane" and change to "The Magnitude of a partial derivative a slope of the tangent plane in the direction of the partial derivative FIXME: Direction Language?"

The Hillside

The direction of the gradient is perpendicular to contour lines The direction of the gradient shows what way is 'uphill' The magnitude of the gradient is proportional to the density of contour lines The components of the gradient are partial derivatives