Is it correct to say: Reading 'increasing' as f(x) > 0 instead of f'(x) > 0?

Monotonicity is decided by the SIGN of the DERIVATIVE, not the sign of the function itself.

Is it correct to say: Applying second-derivative test when f''(x_c) = 0 and concluding 'no extremum'?

f''(x_c) = 0 makes the test INCONCLUSIVE, not negative. Switch to the first-derivative sign-change test.

Is it correct to say: For absolute extrema on [a, b], comparing only critical points and forgetting endpoints?

Always compare f at critical points AND both endpoints. Extrema may occur at a or b.

Is it correct to say: In optimisation, treating the constraint variable as free?

Reduce to ONE variable using the constraint (perimeter/volume) BEFORE differentiating.

Is it correct to say: Related rates: substituting numerical values before differentiating?

Differentiate the general relation first, THEN substitute given instantaneous values. Otherwise you differentiate constants and get zero.

Is it correct to say: Writing slope of normal as −m_t instead of −1/m_t?

Normal slope is the NEGATIVE RECIPROCAL of tangent slope: m_n = −1/m_t.

Application of Derivatives

Tangents, normals, rate of change, monotonicity, maxima and minima, optimisation — NCERT Class 12 Maths Ch 6

📐 12 formulas✏️ 3 examples🎯 6 practice⚖️ 6-8 marks🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→A 5-mark optimisation problem (open box, wire cut into two, largest cylinder in sphere) is asked almost every year.
→For 'increasing/decreasing on an interval' you MUST test the sign of f'(x) on each sub-interval separated by the critical points.
→State the domain restriction (e.g. side length > 0) before applying second-derivative test — CBSE deducts marks if you skip it.
→For tangent/normal questions, first find the point on the curve, then the slope, then use point-slope form.
→Related rates: identify the variable that is changing with time, then differentiate the geometric relation w.r.t. t (not x).

📐 Formulas(12)

Slope of Tangent★ Board fav

m_{t} = \frac{d y}{d x}_{(x_{0}, y_{0})}

Slope of Normal

m_{n} = - \frac{1}{m _{t}}

Equation of Tangent★ Board fav

y - y_{0} = m_{t} (x - x_{0})

Equation of Normal

y - y_{0} = - \frac{1}{m _{t}} (x - x_{0})

Rate of Change★ Board fav

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

Monotonicity Test★ Board fav

f^{'} (x) > 0 on I \Rightarrow f ↑ on I; f^{'} (x) < 0 on I \Rightarrow f ↓ on I

Critical Points

f^{'} (x_{c}) = 0 or f^{'} (x_{c}) undefined

First Derivative Test

f^{'} changes + \to - \Rightarrow local max; - \to + \Rightarrow local min

Second Derivative Test★ Board fav

f^{''} (x_{c}) < 0 \Rightarrow local max; f^{''} (x_{c}) > 0 \Rightarrow local min

Absolute Extrema on [a, b]★ Board fav

max_{[a, b]} f = max {f (crit pts), f (a), f (b)}

Approximation using Differentials

f (x + Δ x) \approx f (x) + f^{'} (x) Δ x

Concavity

f^{''} (x) > 0 \Rightarrow concave up; f^{''} (x) < 0 \Rightarrow concave down

✏️ Solved Examples

1Solved Exampleeasy4 steps

Find the equation of the tangent to the curve y = x² − 4x + 3 at the point where x = 1.

Find y at x = 1 to locate the point of tangency.

y (1) = 1 - 4 + 3 = 0 \Rightarrow (1, 0)

2Solved Exampleboard4 steps

Find the absolute maximum and minimum of f(x) = x³ − 3x² + 4 on [−1, 4].

Find critical points by solving f'(x) = 0.

f^{'} (x) = 3 x^{2} - 6 x = 3 x (x - 2) = 0 \Rightarrow x = 0, x = 2

3Solved ExampleHOTS5 steps

A wire of length 28 m is cut into two pieces; one is bent into a square and the other into a circle. What lengths minimise the total area?

Let x be the length used for the square (side x/4) and 28 − x for the circle (circumference).

side = \frac{x}{4}, 2 π r = 28 - x \Rightarrow r = \frac{28 - x}{2 π}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Reading 'increasing' as f(x) > 0 instead of f'(x) > 0
✓Monotonicity is decided by the SIGN of the DERIVATIVE, not the sign of the function itself.
2
Applying second-derivative test when f''(x_c) = 0 and concluding 'no extremum'
✓f''(x_c) = 0 makes the test INCONCLUSIVE, not negative. Switch to the first-derivative sign-change test.
3
For absolute extrema on [a, b], comparing only critical points and forgetting endpoints
✓Always compare f at critical points AND both endpoints. Extrema may occur at a or b.
4
In optimisation, treating the constraint variable as free
✓Reduce to ONE variable using the constraint (perimeter/volume) BEFORE differentiating.
5
Related rates: substituting numerical values before differentiating
✓Differentiate the general relation first, THEN substitute given instantaneous values. Otherwise you differentiate constants and get zero.
6
Writing slope of normal as −m_t instead of −1/m_t
✓Normal slope is the NEGATIVE RECIPROCAL of tangent slope: m_n = −1/m_t.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the equation of the normal to y = x³ at (1, 1).
Q2
The radius of a circle is increasing at 3 cm/s. How fast is the area increasing when the radius is 10 cm?
Q3
Find the intervals on which f(x) = x³ − 6x² + 9x + 15 is strictly increasing.
Q4
Find two positive numbers whose sum is 24 and whose product is maximum.
Q5
A square sheet of side 18 cm has a small square of side x cut from each corner; the sides are folded to form an open box. Find x that maximises the volume.
Q6
Use differentials to approximate √36.6.

📝 Notes

Applications of Derivatives

Chapter 6 turns the derivative into a decision-making tool. Wherever a quantity changes — the position of a car, the area of a spill, the profit of a factory — the derivative describes the rate, and setting it to zero locates extremes.

Six question-types that dominate the board paper

Tangent / normal: given curve and point, find equations of tangent and normal.
Related rates: given how one quantity changes with time, find how a linked quantity changes.
Monotonicity: find intervals of increase or decrease.
Local extrema: find local maxima and minima of f(x) on an open interval.
Absolute extrema: compare f at critical points and endpoints on a closed interval [a, b].
Optimisation word problem: area, volume, cost, time — build the function, reduce to one variable, differentiate.

A reliable 5-step optimisation recipe

Draw a diagram and name the unknowns.
Write the quantity to be optimised (Q) and the constraint.
Use the constraint to reduce Q to a function of ONE variable, say Q(x).
Differentiate: solve Q'(x) = 0 for critical points, discard values outside the physical domain.
Confirm max/min via Q''(x) or first-derivative sign test, then state the answer with units.

Sanity checks

If your "maximum" is smaller than a value at the boundary, you compared incorrectly.
Areas and lengths must be positive — reject negative roots.
Units on dy/dt should match (units of y) / (units of t). If they do not, you dropped a factor.