Interpretations of derivatives

Looking at derivatives graphically and conceptually.

Contents

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 (x0,f(x0))(x_0, f(x_0)) is

tangent: y−f(x0)=m(x−x0)\text{tangent: }y - f(x_0) = m(x - x_0)

where mm, the gradient of the line, is also called the derivative of f(x)f(x) with respect to xx at x=x0x = x_0, or that

m=f′(x0)m = f'(x_0)

and f′f' is the derivative of ff with respect to xx

From perpendicular lines, the normal line to a point on the graph of y=f(x)y = f(x) has the equation

normal: y−f(x0)=−1m(x−x0)\text{normal: }y - f(x_0) = -\frac{1}{m}(x - x_0)

Since vertical lines do not have defined gradients, for horizontal and vertical tangents or normals use

y=f(x0)x=x0\begin{align*}y &= f(x_0) \\ x &= x_0\end{align*}

Instant rate of change

Derivatives can be interpreted as instantaneous rates of change. For example, speed is

speed=change in distancechange in time=ΔxΔt\text{speed} = \frac{\text{change in distance}}{\text{change in time}} = \frac{\Delta x}{\Delta t}

Over small time intervals, this approaches a derivative

as Δt→0,then ΔxΔt→dx dt\text{as } \Delta t \to 0, \text{then } \frac{\Delta x}{\Delta t} \to \frac{\text{dx}}{{\d t}}

This is discussed with more rigor in limit definition of derivative (HL)

“With respect to”

Differentiating with respect to 3x3x would yield a derivative 13\frac13 as high.

 dy d(3x)=13 dy dx\frac{\d y}{\d(3x)} = \frac{1}{3} \frac{\d y}{\d x}

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 x=x0x = x_0 if and only if (iff) the derivative (with respect to xx) is positive; it is decreasing iff the derivative (with respect to xx) is negative. More on this in extrema and concavity.

increasing: f′(x)>0\text{increasing: }f^\prime(x) > 0
decreasing: f′(x)<0\text{decreasing: }f^\prime(x) < 0

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 f(x0)f(x_0) defined is a necessary but insufficient condition for existence of f′(x0)f^\prime(x_0). More on this in continuity and differentiability (HL).

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