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CBSE Class 12 Maths

Application of Derivatives

Each concept is a short lesson, then questions until you get three right.

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6 concepts, a short lesson each

  • Tangents and rates0/2
  • Increasing functions and extremes0/4

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Derivative as a rate of change

dydx\dfrac{dy}{dx} is the rate at which yy changes with xx; for motion s(t)s(t), velocity v=dsdtv = \dfrac{ds}{dt} and acceleration a=dvdta = \dfrac{dv}{dt}

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Tangents and rates0 of 2

  1. Derivative as a rate of changedydx\dfrac{dy}{dx} is the rate at which yy changes with xx; for motion s(t)s(t), velocity v=dsdtv = \dfrac{ds}{dt} and acceleration a=dvdta = \dfrac{dv}{dt}Start
  2. Related ratesWrite the relation between the quantities, differentiate both sides with respect to tt, then substitute the known rates and valuesStart

Increasing functions and extremes0 of 4

  1. Increasing and decreasing functionsff increases where f′(x)>0f'(x) > 0 and decreases where f′(x)<0f'(x) < 0: find where f′=0f' = 0 and test the sign of f′f' between those pointsStart
  2. Local maxima and minimaAt a critical point f′(c)=0f'(c) = 0: a maximum if f′′(c)<0f''(c) < 0, a minimum if f′′(c)>0f''(c) > 0; or watch f′f' change sign from ++ to −- (maximum) or −- to ++Start
  3. Greatest and least values in problemsWrite the quantity to optimise in one variable using the constraint, find its critical point, and check it is the maximum or minimum asked forStart
  4. Absolute maximum and minimum on a closed intervalOn [a,b][a, b], work out ff at every critical point inside and at both ends: the largest value is the absolute maximum, the smallest the absolute minimumStart

Practise Application of Derivatives

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions