Applications of Derivatives
A derivative is a rate of change
If y depends on x, then dxdy measures how fast y changes as x changes. This is the idea behind every physical rate-of-change problem, from growing areas to moving particles.
Worked example: rate of change of an area
Find the rate of change of the area of a circle with respect to its radius r, when r=4 cm.
Solution: Area is A=πr2, so:
drdA=2πrAt r=4: drdA=2π(4)=8π sq. cm per cm of radius.
Worked example: velocity and acceleration
A particle's position is s=4t3−2t2+3t+7. Find its velocity and acceleration at t=2 seconds.
Solution: Velocity is the first derivative of position, and acceleration is the derivative of velocity:
v=dtds=12t2−4t+3,a=dtdv=24t−4At t=2:
v=12(4)−4(2)+3=43 units/s,a=24(2)−4=44 units/s2Continue learning
Rate-of-change problems and optimization problems both start from the same derivative-as-rate idea: one asks how fast something changes, the other asks where that rate hits zero. The same derivative also drives approximating small changes directly, instead of an exact rate.