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Differentiation revision notes

Every idea in the Differentiation chapter for JEE Main, with the usual mistakes and the fast methods. 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 concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Basic rules and standard derivatives

Derivative from first principles
2× asked, last 2024
f′(x)=lim⁡h→0f(x+h)−f(x)h=lim⁡t→xf(t)−f(x)t−xf'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{t\to x}\frac{f(t)-f(x)}{t-x}
Exists only if the limit is finite. Any limit (G(t)−G(x))/(t−x)(G(t)-G(x))/(t-x) as t→xt\to x is just G′(x)G'(x).
xxyyxxx+hx+hPPh=1h=1: slope 3h=12h=\frac12: 2.5tangent: 2
f=x2f=x^2 at x=1x=1: secant slopes 2+h2+h shrink to the tangent slope f′(1)=2f'(1)=2
  • Quick method. If f(x)−f(y)≥g(x)−g(y)f(x)-f(y)\ge g(x)-g(y) for all x,yx,y, swapping xx and yy gives ≤\le too, so f−gf-g is constant and f′=g′f'=g'. With g=ln⁡x+xg=\ln x+x: f′(x)=1/x+1f'(x)=1/x+1.
Power, exponential and log
2× asked, last 2024
ddxxn=nxn−1,ddxax=axln⁡a,ddxlog⁡ax=1xln⁡a\frac{d}{dx}x^n=nx^{n-1},\qquad \frac{d}{dx}a^x=a^x\ln a,\qquad \frac{d}{dx}\log_a x=\frac{1}{x\ln a}
a>0a>0, a≠1a\ne1; log⁡ax\log_a x needs x>0x>0. (ln⁡∣x∣)′=1/x(\ln|x|)'=1/x for x≠0x\ne0.
  • Not treating a variable base as a constant. Rewrite log⁡g(x)f(x)=ln⁡f/ln⁡g\log_{g(x)}f(x)=\ln f/\ln g, then use the quotient rule. For example log⁡cos⁡xcosec⁡x=−ln⁡sin⁡x/ln⁡cos⁡x\log_{\cos x}\operatorname{cosec}x=-\ln\sin x/\ln\cos x.
Trigonometric derivatives
3× asked, last 2024
(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x,(tan⁡x)′=sec⁡2x,(sec⁡x)′=sec⁡xtan⁡x(\sin x)'=\cos x,\qquad (\cos x)'=-\sin x,\qquad (\tan x)'=\sec^2 x,\qquad (\sec x)'=\sec x\tan x
xx in radians. Co-functions take a minus: (cot⁡x)′=−cosec⁡2x(\cot x)'=-\operatorname{cosec}^2x, and likewise for cosec.
Inverse trigonometric derivatives
4× asked, last 2026
(sin⁡−1x)′=11−x2,(tan⁡−1x)′=11+x2,(sec⁡−1x)′=1∣x∣x2−1(\sin^{-1}x)'=\frac{1}{\sqrt{1-x^2}},\qquad (\tan^{-1}x)'=\frac{1}{1+x^2},\qquad (\sec^{-1}x)'=\frac{1}{|x|\sqrt{x^2-1}}
∣x∣<1|x|<1 for sin⁡−1,cos⁡−1\sin^{-1},\cos^{-1}; ∣x∣>1|x|>1 for sec⁡−1\sec^{-1}. cos⁡−1\cos^{-1}, cot⁡−1\cot^{-1} and cosec−1^{-1} take a minus.
Product and quotient rules
4× asked, last 2024
(uv)′=u′v+uv′,(uv)′=u′v−uv′v2,(uvw)′=u′vw+uv′w+uvw′(uv)'=u'v+uv',\qquad \left(\frac uv\right)'=\frac{u'v-uv'}{v^2},\qquad (uvw)'=u'vw+uv'w+uvw'
Quotient rule needs v≠0v\ne0. Order matters in the numerator: u′v−uv′u'v-uv', not uv′−u′vuv'-u'v.
  • Quick method. If one factor is 0 at x=ax=a, only its derivative term survives: f=u wf=u\,w with u(a)=0u(a)=0 gives f′(a)=u′(a) w(a)f'(a)=u'(a)\,w(a). For f=tan⁡x⋅w(x)f=\tan x\cdot w(x): f′(0)=w(0)f'(0)=w(0); no need to differentiate ww.

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Practise Differentiation

11 more concepts in 5 topics

Chain rule and simplifying first, Inverse trigonometric forms, Implicit and logarithmic differentiation, Parametric differentiation and Higher-order derivatives. 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 before differentiating4×
  • Derivative of an inverse function3×

Inverse trigonometric forms

  • Substitution before differentiating3×
  • Derivative of f with respect to g1×

Implicit and logarithmic differentiation

  • Implicit differentiation7×
  • Logarithmic differentiation5×

Parametric differentiation

  • Parametric first derivative3×
  • Parametric second derivative2×

Higher-order derivatives

  • Second derivative and differential relations6×
  • Functions defined by their own derivatives5×

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