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Differentiation formula sheet

Every formula in the Differentiation chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 16 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.

Basic rules and standard derivatives

First principles
2× asked, last 2024
f′(x)=lim⁡h→0f(x+h)−f(x)hf′(x)=lim⁡t→xf(t)−f(x)t−x\begin{aligned}&f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h\\&\phantom{f^{\prime}(x)}=\lim_{t\to x}\frac{f(t)-f(x)}{t-x}\end{aligned}
Finite limit only. Any limit (G(t)−G(x))/(t−x)(G(t)-G(x))/(t-x) as t→xt\to x is G′(x)G^{\prime}(x).
Power, exponential and log
2× asked, last 2024
(xn)′=nxn−1,(ax)′=axln⁡a(log⁡ax)′=1xln⁡a\begin{aligned}&(x^n)^{\prime}=nx^{n-1},\qquad (a^x)^{\prime}=a^x\ln a\\&(\log_ax)^{\prime}=\frac1{x\ln a}\end{aligned}
(ln⁡∣x∣)′=1/x(\ln|x|)^{\prime}=1/x. Variable base: log⁡gf=ln⁡f/ln⁡g\log_gf=\ln f/\ln g, then the quotient rule.
Trigonometric
3× asked, last 2024
(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x(tan⁡x)′=sec⁡2x,(sec⁡x)′=sec⁡xtan⁡x\begin{aligned}&(\sin x)^{\prime}=\cos x,\qquad (\cos x)^{\prime}=-\sin x\\&(\tan x)^{\prime}=\sec^2x,\qquad (\sec x)^{\prime}=\sec x\tan x\end{aligned}
Radians. Co-functions take a minus: (cot⁡x)′=−cosec⁡2x(\cot x)^{\prime}=-\operatorname{cosec}^2x, (cosec⁡x)′=−cosec⁡xcot⁡x(\operatorname{cosec}x)^{\prime}=-\operatorname{cosec}x\cot x.
Inverse trigonometric
4× asked, last 2026
(sin⁡−1x)′=11−x2,(tan⁡−1x)′=11+x2(sec⁡−1x)′=1∣x∣x2−1\begin{aligned}&(\sin^{-1}x)^{\prime}=\frac1{\sqrt{1-x^2}},\qquad (\tan^{-1}x)^{\prime}=\frac1{1+x^2}\\&(\sec^{-1}x)^{\prime}=\frac1{|x|\sqrt{x^2-1}}\end{aligned}
cos⁡−1\cos^{-1}, cot⁡−1\cot^{-1}, cosec⁡−1\operatorname{cosec}^{-1}: same with a minus. ∣x∣<1|x|<1 for sin⁡−1\sin^{-1}, ∣x∣>1|x|>1 for sec⁡−1\sec^{-1}.
Product and quotient
4× asked, last 2024
(uv)′=u′v+uv′,(uv)′=u′v−uv′v2(uv)^{\prime}=u^{\prime}v+uv^{\prime},\qquad \left(\frac uv\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^2}
(uvw)′=u′vw+uv′w+uvw′(uvw)^{\prime}=u^{\prime}vw+uv^{\prime}w+uvw^{\prime}. If u(a)=0u(a)=0: (uw)′(a)=u′(a) w(a)(uw)^{\prime}(a)=u^{\prime}(a)\,w(a).

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Practise Differentiation

11 more formulas in 4 topics

Chain rule and simplifying first, Inverse trigonometric forms, Implicit and logarithmic and Parametric and higher order. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Chain rule and simplifying first

  • Chain rule6×
  • Simplify first4×
  • Inverse function3×

Inverse trigonometric forms

  • Double-angle forms3×
  • Which substitution

Implicit and logarithmic

  • Implicit differentiation7×
  • Logarithmic differentiation5×

Parametric and higher order

  • Parametric first derivative3×
  • Parametric second derivative2×
  • Differential relations6×
  • Defined by own derivatives5×

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About 40 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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Version 1, October 2026. Free to share for study, as it is. Please don't sell it or remove the PrepWiser name.

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