Knowledge organisers

Application of Derivatives: CBSE Class 12 Maths knowledge organiser

Everything to know about application of derivatives on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Increasing function: f′(x) > 0 on an interval means f is strictly increasing there.
Critical point: A point where f′(c) = 0 or f′(c) does not exist.
Local maximum: A point higher than all nearby points: f′ changes from + to −.

Key formulas

Rate of change	\(\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}\)
\(f'(x)>0\) on \((a,b)\): strictly increasing; \(f'(x)<0\): strictly decreasing
Critical point: \(f'(c)=0\) or \(f'(c)\) undefined
First derivative test: \(f'\) changes + → − (max), − → + (min), no change (neither)
Second derivative test	\(f'(c)=0:\) \(f''(c)<0\text{ max},\) \(f''(c)>0\text{ min},\) \(f''(c)=0\text{ test fails}\)
Absolute max/min on \([a,b]\): compare \(f\) at critical points and at \(a\), \(b\)

Worked example

A spherical balloon is inflated at \(100\ \text{cm}^3/\text{s}\). When \(r = 5\) cm: \(\dfrac{dV}{dt} = 4\pi r^2\dfrac{dr}{dt} \Rightarrow \dfrac{dr}{dt} = \dfrac{100}{100\pi} = \dfrac1\pi\ \text{cm/s}\).

Common mistakes

Substituting the particular value before differentiating in a rate problem.
Forgetting end points when finding absolute extrema; confusing a local maximum with the absolute maximum.
Ignoring points where \(f'\) is undefined (e.g. \(x^{2/3}\) at \(0\)).
Not justifying that the stationary point is a maximum/minimum — this carries a mark.

You should be able to…

Find a rate of change, find where a function increases or decreases and locate turning points.
Classify local maxima and minima with a derivative test and find absolute extrema on a closed interval.
Write full-marks optimisation and case-study answers: function set up, derivative, test, and the value that was asked for.
Handle monotonicity with parameters or trig functions and the harder optimisation problems.

The printable sheet

Application of Derivatives knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Application of Derivatives knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers