Calculus: Derivatives and Motion

The derivative measures the rate of change of a function at a particular input. $f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$. It is the slope of the tangent line at $x=a$.

A statement identifying the input value, the quantity changing, its direction, and its units. For example, $P'(4)=12$ means that at $t=4$, population is increasing at 12 people per time unit.

Output units divided by input units. $\displaystyle [f'(x)]=\frac{\text{output units}}{\text{input units}}$. For example, $dV/dt$ may be measured in liters per minute.

If $f'(a)>0$, the instantaneous rate is positive at $a$. If $f'(x)>0$ throughout an interval, the function is increasing there.

If $f'(a)<0$, the instantaneous rate is negative at $a$. If $f'(x)<0$ throughout an interval, the function is decreasing there.

If $f'(a)=0$, the tangent is horizontal at $a$. This does not necessarily mean there is a maximum, minimum, or direction change.

The instantaneous rate of change of the first derivative. $f''(x)=\frac{d}{dx}f'(x)$. Units are output units divided by squared input units.

Signed location relative to a chosen origin, represented by $s(t)$. Its sign indicates location, not necessarily direction of motion.

The signed change in position over an interval. $\Delta s=s(b)-s(a)$. It may be positive, negative, or zero.

The entire length of the path regardless of direction. $D=\int_a^b|v(t)|\,dt$. Alternatively, sum absolute position changes between direction changes.

The instantaneous rate of change of position. $v(t)=s'(t)=\frac{ds}{dt}$. Its sign indicates direction of motion.

The magnitude of velocity. $\text{Speed}=|v(t)|$. Speed is nonnegative.

The instantaneous rate of change of velocity. $a(t)=v'(t)=s''(t)$. If position is in meters and time in seconds, acceleration is in $\mathrm{m/s^2}$.

When $v(t)=0$. Being at rest does not necessarily mean the particle changes direction.

If $v(t)>0$, the particle moves in the positive coordinate direction and its position is increasing.

If $v(t)<0$, the particle moves in the negative coordinate direction and its position is decreasing.

When speed is increasing. For nonzero velocity, velocity and acceleration have the same sign: $v(t)a(t)>0$.

When speed is decreasing. For nonzero velocity, velocity and acceleration have opposite signs: $v(t)a(t)<0$.

Displacement divided by elapsed time. $v_{\mathrm{avg}}=\frac{s(b)-s(a)}{b-a}$. Average velocity is signed.

Total distance divided by elapsed time. $\text{Average speed}=\frac{D}{b-a}$. It is nonnegative.

A derivative describing how a real-world quantity changes with another. Examples: $dV/dt$ for volume, $dT/dt$ for temperature, and $dP/dt$ for population.

Problems involving changing quantities linked by an equation. Differentiate the relationship with respect to time to relate their rates.

Differentiating an equation without isolating the dependent variable. Apply the chain rule: $\frac{d}{dt}(y^2)=2y\frac{dy}{dt}$.

If $y=f(x)$ and $x=x(t)$, then $\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}$.

A derivative supplied by the problem. For example, $\frac{dr}{dt}=3\ \mathrm{cm/s}$ means the radius is increasing at 3 centimeters per second.

The derivative to be calculated from the relationship and known rates, such as $dV/dt$ for a changing sphere.

An equation linking geometric quantities, such as $x^2+y^2=L^2$, $A=\pi r^2$, or $V=\frac13\pi r^2h$. Differentiate with respect to time for related rates.

A quantity that remains unchanged over time. If $L$ is constant, then $\frac{dL}{dt}=0$.

A differentiable function is approximately linear near a point. $f(a+h)\approx f(a)+f'(a)h$ for sufficiently small $h$.

The tangent-line function used to approximate $f$ near $a$. $L(x)=f(a)+f'(a)(x-a)$.

Estimating a nearby function value with the tangent line: $f(x)\approx L(x)=f(a)+f'(a)(x-a)$.

The linear estimate of a small output change. For $y=f(x)$, $dy=f'(x)\,dx$.

The chosen change in the independent variable. $dx=\Delta x$ when estimating the change from an initial input.

The linear estimate of the dependent-variable change. $dy=f'(x)\,dx\approx\Delta y$ for sufficiently small $dx$.

The exact output change. $\Delta y=f(x+\Delta x)-f(x)$.

The difference between an actual value and its approximation. $E=f(x)-L(x)$; absolute error is $|E|=|f(x)-L(x)|$.

If $f''(x)>0$ throughout the relevant interval, the graph lies above its tangent line. Therefore $L(x)\le f(x)$ and linearization underestimates or is exact at tangency.

If $f''(x)<0$ throughout the relevant interval, the graph lies below its tangent line. Therefore $L(x)\ge f(x)$ and linearization overestimates or is exact at tangency.

A limiting expression whose apparent form does not uniquely determine its value. Examples include $0/0$ and $\infty/\infty$.

A quotient limit where numerator and denominator both approach zero. This alone does not determine the quotient limit.

A quotient limit where numerator and denominator both become unbounded in magnitude. The quotient may have several possible limiting behaviors.

Under the rule's hypotheses, for applicable $0/0$ or $\infty/\infty$ limits, $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$, provided the derivative quotient has a finite or infinite limit.

When applicable, replace $\frac{f(x)}{g(x)}$ with $\frac{f'(x)}{g'(x)}$ for evaluating the limit. This is not the quotient rule.

When the derivative quotient again produces an applicable $0/0$ or $\infty/\infty$ form and the rule's conditions still hold. Recheck after every application.

A complete contextual statement including the input, quantity, direction, rate, and units. If $V'(5)=-8$, with liters and minutes, then at 5 minutes the volume is decreasing at 8 liters per minute.

Checking that units are consistent. For a sphere, $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$ has units $\mathrm{cm^2}\cdot\mathrm{cm/s}=\mathrm{cm^3/s}$.