PREPWISER.
Practise online

Applications of Derivatives revision notes

Every idea in the Applications of Derivatives chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 22 and download the PDF.

Sign in to read all 22Download PDF

Read the first part free. Sign in free with Google to read the rest and download the PDF: 11 A4 pages.11 A4 pages, ready to print.

How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Rate of change

Related rates (chain rule in t)
4× asked, last 2025
dydt=dydx⋅dxdt,dVdt=4πr2drdt,dSdt=8πrdrdt\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt},\qquad \frac{dV}{dt}=4\pi r^2\frac{dr}{dt},\qquad \frac{dS}{dt}=8\pi r\frac{dr}{dt}
Write the quantity in one variable, differentiate in tt, then put in the instant's values. A decrease is a negative rate.
  • Not the inner radius for a shell of uniform thickness hh on a core of radius rr. The volume changes at the outer surface: dV/dt=4π(r+h)2 dh/dtdV/dt=4\pi(r+h)^2\,dh/dt. Find r+hr+h, then subtract hh.
Cone being filled (vertex down)
2× asked, last 2022
r=htan⁡α,V=π3h3tan⁡2α,Swet=πrl=πh2tan⁡αsec⁡αr=h\tan\alpha,\qquad V=\frac{\pi}{3}h^3\tan^2\alpha,\qquad S_{\text{wet}}=\pi r l=\pi h^2\tan\alpha\sec\alpha
α\alpha = semi-vertical angle, fixed. Remove rr by similar triangles first; get dh/dtdh/dt once from dV/dtdV/dt and reuse it.
3514h=10h=10r=2r=2α\alphatan⁡α=735=15\tan\alpha=\frac7{35}=\frac15dhdt=14π\frac{dh}{dt}=\frac1{4\pi}dSdt=265\frac{dS}{dt}=\frac{\sqrt{26}}5
To scale: r=htan⁡αr=h\tan\alpha with tan⁡α=735\tan\alpha=\frac7{35}, so at h=10h=10 the surface has r=2r=2
  • Not dS/dt=c dh/dtdS/dt=c\,dh/dt. The wet area is ch2ch^2, so dS/dt=2ch dh/dtdS/dt=2ch\,dh/dt. Height 35, diameter 14, inflow 1 cm³/s, h=10h=10: dh/dt=1/(4π)dh/dt=1/(4\pi) and dS/dt=26/5dS/dt=\sqrt{26}/5.

Practise Applications of Derivatives questions as you read. Free, and you can start without signing up.

Practise Applications of Derivatives

20 more concepts in 5 topics

Increasing and decreasing functions, Counting real roots, Maxima and minima, Polynomials with given extrema and Optimisation. Read the first part free. Sign in free with Google to read the rest and download the PDF.

Sign in to read the restYou come straight back to this spot.Download the PDF

Increasing and decreasing functions

  • Sign of f' and monotonicity13×
  • Monotonic on a whole interval (parameter)4×
  • Monotonicity of composites6×
  • Comparing values with monotonicity2×

Counting real roots

  • Roots from monotonic pieces6×
  • Rolle's theorem: zeros of derivatives8×

Maxima and minima

  • Critical points5×
  • First-derivative test9×
  • Counting local extrema6×
  • Second-derivative test and inflection3×
  • Absolute extrema on a closed interval9×
  • Extrema at corners and joins4×

Polynomials with given extrema

  • Cubic from its critical points10×
  • Polynomial from a limit and extrema4×

Optimisation

  • Setting up an optimisation9×
  • Cutting a wire into two figures4×
  • Standard optimisation results
  • AM-GM and Cauchy shortcuts4×
  • Least distance to a curve6×
  • Max and min by substitution3×

Practise Applications of Derivatives next

About 155 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Applications of Derivatives

Also free on PrepWiser

Spotted a mistake? Write to support@prepwiser.in and we will fix it in the next version.

Version 1, October 2026. Free to share for study, as it is. Please don't sell it or remove the PrepWiser name.

PractiseSign in to read all 22Download PDF
Free with Google, takes a few seconds11 pages, free