CBSE · Class 12 · Mathematics · Chapter 6

Application of Derivatives — Formula Sheet

Board Formulas

12 formulas

1.Slope of Tangent★
$m_{t} = \frac{d y}{d x}_{(x_{0}, y_{0})}$
Evaluate the derivative at the point of tangency. Vertical tangent when dy/dx is infinite.
2.Slope of Normal
$m_{n} = - \frac{1}{m _{t}}$
Normal is perpendicular to tangent, so product of slopes = −1. Horizontal tangent gives vertical normal (undefined slope).
3.Equation of Tangent★
$y - y_{0} = m_{t} (x - x_{0})$
Point-slope form at the point of tangency (x₀, y₀).
4.Equation of Normal
$y - y_{0} = - \frac{1}{m _{t}} (x - x_{0})$
Same point, reciprocal-negative slope. If m_t = 0, normal is x = x₀ (vertical).
5.Rate of Change★
$\frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}$
Related-rates workhorse. Both y and x are functions of time; chain rule links them.
6.Monotonicity Test★
$f^{'} (x) > 0 on I \Rightarrow f ↑ on I; f^{'} (x) < 0 on I \Rightarrow f ↓ on I$
Test on the OPEN interval. Boundary points can be included if f is continuous there.
7.Critical Points
$f^{'} (x_{c}) = 0 or f^{'} (x_{c}) undefined$
Candidates for local maxima/minima. Also check ENDPOINTS on a closed interval for absolute extrema.
8.First Derivative Test
$f^{'} changes + \to - \Rightarrow local max; - \to + \Rightarrow local min$
Sign chart of f' across the critical point. Works even when f'' = 0.
9.Second Derivative Test★
$f^{''} (x_{c}) < 0 \Rightarrow local max; f^{''} (x_{c}) > 0 \Rightarrow local min$
Inconclusive when f''(x_c) = 0 — fall back to first-derivative test.
10.Absolute Extrema on [a, b]★
$max_{[a, b]} f = max {f (crit pts), f (a), f (b)}$
Compare f at all critical points inside (a, b) PLUS the two endpoints. Same for minimum.
11.Approximation using Differentials
$f (x + Δ x) \approx f (x) + f^{'} (x) Δ x$
Linear approximation. Used to estimate √25.3 ≈ 5 + (0.3)/(2·5) = 5.03.
12.Concavity
$f^{''} (x) > 0 \Rightarrow concave up; f^{''} (x) < 0 \Rightarrow concave down$
Points where f''(x) changes sign are points of inflection.