Interpretations of derivatives
Looking at derivatives graphically and conceptually.
Contents
- Secant and tangent lines
- Instant rate of change
- “With respect to”
- Increasing and decreasing functions
- Derivatives as functions
Secant and tangent lines
On a circle, a secant line crosses the circle at two points, while a tangent line only crosses once.
For graphs, it’s a very similar idea. Imagine a smooth curve being replaced with a series of jagged line segments. Each line segment can be extended into a secant or tangent line. It is a tangent line if the line segment intersects the graph at exactly one point; It is a secant line if it crosses the graph at two points.
The point-gradient equation of a tangent line through is
where , the gradient of the line, is also called the derivative of with respect to at , or that
and is the derivative of with respect to
From perpendicular lines, the normal line to a point on the graph of has the equation
Since vertical lines do not have defined gradients, for horizontal and vertical tangents or normals use
Instant rate of change
Derivatives can be interpreted as instantaneous rates of change. For example, speed is
Over small time intervals, this approaches a derivative
This is discussed with more rigor in limit definition of derivative (HL)
“With respect to”
Differentiating with respect to would yield a derivative as high.
Intuitively, derivative is like gradient, and dividing by a larger number decreases the slope.
So derivatives have analogous properties as fractions. This is further discussed in chain rule and related rates (HL).
Increasing and decreasing functions
A function is increasing at if and only if (iff) the derivative (with respect to ) is positive; it is decreasing iff the derivative (with respect to ) is negative. More on this in extrema and concavity.
Derivatives as functions
Derivatives satisfy the definition of functions of one input means at most one output. This allows our knowledge on functions to be carried over to calculus. The domain of a derivative function is a subset of the domain of the original function. This means having defined is a necessary but insufficient condition for existence of . More on this in continuity and differentiability (HL).