The Derivative and Rates of Change

Finding the line tangent to a curve at a point is geometry; finding the velocity of a falling ball at an instant is physics. Both reduce to evaluating a limit of the form

That limit is the derivative. Read one way it is the slope of a tangent line; read another it is an instantaneous rate of change. Letting the base point vary turns it from a number attached to one point into a function in its own right.

The tangent problem

A secant line through two points of a curve has a slope we can compute with algebra. The tangent line touches at a single point, so the slope formula $\frac{y_2-y_1}{x_2-x_1}$ has nothing to work with. The way around is to compute the slope of a secant and then slide one intersection point toward the other.

Let $C$ be the graph of $y=f(x)$ and fix a point $P(a,f(a))$ on it. A nearby point $Q(x,f(x))$ with $x\ne a$ determines the secant line $PQ$, whose slope is

As $x\to a$, the point $Q$ slides along the curve toward $P$ and the secant pivots. If the secant slopes approach a single number $m$, that number is the slope of the tangent, and the tangent is the limiting position of the secant.

Example. For the parabola $y=x^2$ at $P(1,1)$, take $a=1$ and $f(x)=x^2$:

The point-slope form gives the tangent $y-1=2(x-1)$, that is, $y=2x-1$. Zooming in on $(1,1)$ makes the parabola indistinguishable from this line, which is the geometric content of the tangent being a local approximation to the curve.

The $h$ form

Setting $h=x-a$ (so $x=a+h$ and $h\to 0$ as $x\to a$) rewrites the same slope with the increment $h$ as the variable:

The quotient $\frac{f(a+h)-f(a)}{h}$ is the difference quotient: rise over run for the secant joining $P(a,f(a))$ to $Q(a+h,f(a+h))$.

Example. For $y=3/x$ at $(3,1)$, with $f(x)=3/x$:

The tangent at $(3,1)$ is $y-1=-\frac{1}{3}(x-3)$, or $x+3y-6=0$.

Velocity and the rate of change

Suppose an object moves along a line with position $s=f(t)$, the directed distance from the origin at time $t$. Over the interval from $t=a$ to $t=a+h$ the displacement is $f(a+h)-f(a)$, so the average velocity is

exactly the slope of the secant. Letting $h\to 0$ shrinks the interval to the instant $t=a$ and defines the instantaneous velocity:

The tangent-slope limit and the velocity limit are identical in form, so they are computed the same way. This is the general pattern: whenever $y=f(x)$, the quantity

measures how fast $y$ changes with respect to $x$ at $x=a$.

Example. A ball dropped from a $450$ m tower falls $s=f(t)=4.9t^2$ metres in $t$ seconds. Its velocity at a general time $t$ is

After $5$ s the velocity is $v(5)=49$ m/s. The ball lands when $4.9t^2=450$, i.e. $t=\sqrt{450/4.9}\approx 9.6$ s, so its impact speed is $v(\sqrt{450/4.9})\approx 94$ m/s.

Both forms are used; the $h$ form usually leads to simpler algebra. The tangent line to $y=f(x)$ at $(a,f(a))$ therefore has the compact equation

Example. For $f(x)=x^2-8x+9$,

At $a=3$ the slope is $f^{\prime }(3)=-2$, so the tangent at $(3,-6)$ is $y=-2x$.

Because average rate of change is the difference quotient and instantaneous rate of change is its limit, the derivative is the single object underneath several names.

The units follow the ratio: $f^{\prime }(a)$ carries the units of $y$ divided by the units of $x$.

The derivative as a function

Fixing $a$ produced a number. Letting the base point vary produces a function. Replace the constant $a$ by the variable $x$:

For each $x$ where this limit exists we assign the value $f^{\prime }(x)$, so $f^{\prime }$ is a new function derived from $f$. Its value at $x$ is the slope of the tangent to $f$ at $(x,f(x))$. The domain of $f^{\prime }$ is the set of $x$ where the limit exists, and it may be smaller than the domain of $f$.

Example. For $f(x)=x^3-x$, treating $x$ as constant and $h$ as the limit variable,

Where $f$ has a horizontal tangent, $f^{\prime }(x)=0$; where $f$ rises, $f^{\prime }(x)>0$; where $f$ falls, $f^{\prime }(x)<0$. The derivative graph therefore encodes the shape of $f$: its zeros mark the horizontal tangents, its sign marks the rise and fall.

Example. For $f(x)=\sqrt{x}$, rationalizing the numerator,

Here the domain shrinks: $f$ is defined on $[0,\infty )$ but $f^{\prime }$ only on $(0,\infty )$, because the tangent at the origin is vertical.

Notation

Writing $y=f(x)$, the derivative goes by several equivalent symbols. The Leibniz symbol $\mathbf{d}y/\mathbf{d}x$ is a synonym for $f^{\prime }(x)$; for now it is one indivisible piece of notation, not a genuine ratio of two quantities.

To evaluate a Leibniz derivative at a specific number, write ${\frac{\mathbf{d}y}{\mathbf{d}x}|}_{x=a}$, which means $f^{\prime }(a)$.

Differentiability implies continuity

Differentiability is a stronger condition than continuity: a smooth-enough function is automatically unbroken, though the converse fails.

The proof connects the given (the derivative exists) to the goal (the function value is the limit) by inserting the difference quotient. For $x\ne a$,

so by the product law for limits,

Therefore ${\lim }_{x\to a}f(x)=f(a)$, which is continuity at $a$.

The converse is false. The function $f(x)=|x|$ is continuous everywhere but not differentiable at $0$, because the one-sided limits of the difference quotient disagree:

Failure modes of differentiability

A function can fail to be differentiable at $a$ in three geometrically distinct ways.

• A corner. The graph changes direction abruptly, so the left and right difference quotients approach different slopes (as for $|x|$ at $0$). No single tangent line exists.
• A discontinuity. If $f$ is not continuous at $a$, the previous theorem (in contrapositive form) forbids differentiability. A jump discontinuity is the standard case.
• A vertical tangent. The graph is continuous and the tangent lines get steeper without bound, so ${\lim }_{x\to a}|f^{\prime }(x)|=\infty$ and the slope is not a finite number (as for $\sqrt[3]{x}$ at $0$).

Differentiability at $a$ has a hands-on test: zoom in on $(a,f(a))$. If the graph straightens into a line, $f$ is differentiable there; a corner or a break persists at any magnification.

Higher derivatives

The derivative $f^{\prime }$ is itself a function, so it may have a derivative. The second derivative $f^{\prime \prime }=(f^{\prime }{)}^{\prime }$ measures the rate of change of the slope:

Example. For $f(x)=x^3-x$ we found $f^{\prime }(x)=3x^2-1$, so

Continuing, $f^{\prime \prime \prime }(x)=6$ and $f^{(4)}(x)=0$. In general the $n$th derivative $f^{(n)}$ is obtained by differentiating $n$ times, written ${\mathbf{d}}^{n}y/\mathbf{d}{x}^n$.

For a position function $s=s(t)$, the derivatives are the standard chain of motion quantities.

A large jerk is a sudden change in acceleration, felt as an abrupt lurch in a vehicle. The second derivative also measures concavity and inflection, and the full tower $f^{(n)}$ supplies the coefficients of the Taylor series that represent a function as an infinite polynomial.