“Sad face, happy face”
The second derivative test
Negative second derivative means concave down, a sad face, so a maximum. Positive means concave up, a happy face, so a minimum.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Applications of Derivatives formulas
Derivatives stop being algebra here and start answering questions: how fast, how steep, and what is the best. Maxima and minima carry the most marks, and the second derivative test settles them quickly.
15 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Rate of change
| What | Formula | Symbols and notes |
|---|---|---|
| Rate of change | dy/dx is the rate of y with respect to x | related rates chain through a common variable |
| Marginal cost | MC = dC/dx | the cost of producing one more unit |
| Approximation | f(x + Δx) ≈ f(x) + f'(x)Δx | used for small changes and errors |
| Percentage error | (Δy/y) × 100 ≈ (f'(x)Δx/y) × 100 | how an input error propagates |
Tangents and normals
| What | Formula | Symbols and notes |
|---|---|---|
| Slope of the tangent | m = dy/dx at the point | the derivative evaluated there |
| Equation of the tangent | y − y₁ = m(x − x₁) | the point-slope form |
| Slope of the normal | −1/m | perpendicular to the tangent |
| Horizontal tangent | dy/dx = 0 | and the tangent is vertical when dy/dx is undefined |
Increasing, decreasing and extrema
| What | Formula | Symbols and notes |
|---|---|---|
| Increasing | f'(x) > 0 on the interval | the curve rises left to right |
| Decreasing | f'(x) < 0 on the interval | the curve falls |
| Critical point | f'(x) = 0 or undefined | the only candidates for a maximum or minimum |
| First derivative test | sign of f' changes plus to minus at a maximum | minus to plus at a minimum |
| Second derivative test | f''(x) < 0 maximum, f''(x) > 0 minimum | inconclusive when f''(x) = 0 |
| Point of inflexion | f''(x) = 0 and the concavity changes | the curve switches from cup to cap |
| Absolute extrema | check the critical points and both endpoints | on a closed interval |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Critical first, test second”
Finding extrema
Set the first derivative to zero to find candidates, then use the second derivative or a sign change to decide which is which.
“Do not forget the ends”
Absolute maximum on a closed interval
The largest value may sit at an endpoint where the derivative is not zero at all. Always compare the endpoints too.
Where marks go
The mistakes that cost the most.
- Stopping at f'(x) = 0 and calling it a maximum without testing which it is.
- Ignoring the endpoints when asked for the absolute maximum or minimum on a closed interval.
- Using the second derivative test when f''(x) = 0, where it gives no answer.
- Taking the normal's slope as m rather than −1/m.
- Confusing a point of inflexion with an extremum. At an inflexion the curve keeps going, it only changes its bend.
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