xSkill: notes, lectures and practice for CBSE and CSE

1.1 Rates of change When a quantity y depends on time through x, the chain rule turns the rate in x into the rate in y. Differentiate the relation between the variables with respect to t, then substitute the instant values. The sign of the derivative tells whether the quantity grows or shrinks at that moment. The same rule covers areas, volumes and distances: write the formula, differentiate both sides with respect to time, and substitute the given rate as a number with its unit.

\frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}

Chain rule for related rates

\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t} for a sphere

Volume rate of a sphere in terms of the radius rate

1.2 Increasing and decreasing functions A differentiable function rises on an interval when its derivative is positive there and falls when the derivative is negative. The critical points, where the derivative is zero or fails to exist, split the domain into intervals. Test one point per interval, then state the intervals of increase and decrease with open or closed endpoints as the question requires.

f^{'} (x) > 0 on I \Rightarrow f strictly increasing on I

Positive derivative means rising

f^{'} (x) < 0 on I \Rightarrow f strictly decreasing on I

Negative derivative means falling

1.3 Maxima, minima and optimisation A local maximum occurs where the derivative changes from positive to negative, and a local minimum where it changes from negative to positive. The second derivative test checks the sign at a critical point: negative means a local maximum, positive means a local minimum, zero is inconclusive. For a closed interval, compare the values at the critical points with the endpoint values to get the absolute maximum and minimum. Word problems on maximum area, volume or minimum cost follow the same three steps: build the function, differentiate, and test the critical point.

f^{'} (c) = 0, f^{''} (c) < 0 \Rightarrow local maximum at c

Second derivative test for a maximum

f^{'} (c) = 0, f^{''} (c) > 0 \Rightarrow local minimum at c

Second derivative test for a minimum

m = f^{'} (x_{0}), m_{normal} = - \frac{1}{f ^{'} ( x _{0} )}

Slopes of the tangent and the normal at a point

1.4 Standard results for quick reference For a sphere, volume rate equals four pi r squared times radius rate, and surface area rate equals eight pi r times radius rate. For a circle, area rate equals two pi r times radius rate. For a cube, volume rate equals three a squared times side rate. A function increases where its derivative is positive and decreases where it is negative. At a stationary point, a negative second derivative gives a local maximum and a positive one gives a local minimum. For a closed interval, the absolute extrema are among the critical points and the endpoints.

Exam

In optimisation problems, state the quantity to be maximised or minimised, write it as a function of one variable using the constraint, and mention the domain. The derivative alone does not finish the answer; the test step earns its own mark.