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Applications of Derivatives formula sheet

Every formula in the Applications of Derivatives chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 24 and download the PDF.

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

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4× asked, last 2025
dydt=dydx⋅dxdtsphere: dVdt=4πr2drdt,dSdt=8πrdrdt\begin{aligned}&\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}\\&\text{sphere: }\frac{dV}{dt}=4\pi r^2\frac{dr}{dt},\quad \frac{dS}{dt}=8\pi r\frac{dr}{dt}\end{aligned}
One variable, differentiate in tt, then put in values; a decrease is negative. Shell: use the outer radius.
Cone being filled
2× asked, last 2022
r=htan⁡α,V=π3h3tan⁡2αSwet=πrl=πh2tan⁡αsec⁡α\begin{aligned}&r=h\tan\alpha,\qquad V=\frac\pi3h^3\tan^2\alpha\\&S_{\text{wet}}=\pi rl=\pi h^2\tan\alpha\sec\alpha\end{aligned}
α\alpha fixed; dS/dt=2ch dh/dtdS/dt=2ch\,dh/dt. H=35H=35, D=14D=14, 1 cm³/s, h=10h=10: dh/dt=1/(4π)dh/dt=1/(4\pi), dS/dt=26/5dS/dt=\sqrt{26}/5.

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22 more formulas 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.

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Increasing and decreasing functions

  • Sign of f′13×
  • Proving f′ > 0
  • Monotonic for a parameter4×
  • Composites6×
  • Comparing values2×

Counting real roots

  • Roots from monotonic pieces6×
  • Cubic equal to k
  • Rolle's theorem8×

Maxima and minima

  • Critical points5×
  • First-derivative test9×
  • Counting local extrema6×
  • Second-derivative test3×
  • Absolute extrema on [a, b]9×
  • Corners and joins4×

Polynomials with given extrema

  • Cubic from its critical points10×
  • Polynomial from a limit4×

Optimisation

  • Setting up9×
  • Cutting a wire4×
  • Standard results
  • AM-GM and Cauchy4×
  • Least distance to a curve6×
  • By substitution3×

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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.

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