Q.A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.

The derivative $f'(x)$ of a function carries two equivalent readings, and this chapter leans on both throughout.

As a slope. For the curve $y=f(x)$, the slope of the chord joining $(x,f(x))$ and $(x+h,f(x+h))$ is the Newton quotient

Taking $h\to0$ gives the slope of the curve at $(x,f(x))$ itself:

If $\theta$ is the angle the tangent makes with the positive $x$-axis (measured anticlockwise), then $f'(x)=\tan\theta$.

As a rate of change. $f'(x)=\dfrac{dy}{dx}$ is also the instantaneous rate of change of $y$ with respect to $x$; over an interval $[a,b]$ the average rate of change is the ordinary difference quotient $\dfrac{f(b)-f(a)}{b-a}$ (a chord slope), while the derivative at a single point is the instantaneous rate.

Motion along a line. If $s=f(t)$ is the position of an object at time $t$ (measured from a fixed origin, positive direction = forward):

• Speed $=|v(t)|=\left|\dfrac{ds}{dt}\right|$ — always non-negative, regardless of direction.
• $v(t)=0$: the particle is momentarily at rest.
• $v(t)>0$: moving forward; $v(t)<0$: moving backward.
• The particle changes direction exactly where $v(t)$ changes sign (not merely where it is zero — the sign must flip on either side).
• If the particle reverses direction at time $t_c\in(t_1,t_2)$, the total distance travelled from $t_1$ to $t_2$ is $|s(t_c)-s(t_1)|+|s(t_2)-s(t_c)|$ — NOT simply $|s(t_2)-s(t_1)|$, since backtracking would otherwise cancel out.
• Near Earth's surface a freely falling body has constant acceleration $g\approx9.8\ \text{m/s}^2$ ($32\ \text{ft/s}^2$), giving $a=-g,\ v=-gt+v_0,\ s=-\tfrac12gt^2+v_0t+s_0$.

Related rates. A related-rates problem links two or more time-varying quantities through a single equation (areas, volumes, distances, …); differentiating that equation with respect to time $t$ (using the chain rule on every quantity that depends on $t$) produces an equation relating the rates $\dfrac{d(\cdot)}{dt}$, which is then solved for the unknown rate at the instant specified. The recurring workflow is: (1) write the governing geometric/physical relation, (2) differentiate both sides with respect to $t$, (3) substitute the known values (and any values found from the constraint at that instant), (4) solve for the required rate.