Application of Derivatives
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Rates of Change
Part of: Rates and Rising Graphs
A derivative dy/dx measures how fast y moves per unit of x, and chained rates divide through time. Circle area grows with radius, cube surface follows volume, rectangles balance two moving sides, and marginal cost or revenue is the cost or revenue slope at a production level.
Why it matters. Engineers track growth rates and firms price units through these exact computations.
Formulas
- Circle area rate is 2πr times radius rate
- Chain volume rate into surface rate
- Marginal means differentiate, then plug the level
Key words
- Rate of change
- : A derivative measuring speed of one quantity against another.
- Marginal cost
- : The cost slope at a given production level.
- Marginal revenue
- : The revenue slope at a given sales level.
- Related rates
- : Linked time rates connected by the chain rule.
Try it in two minutes
- Differentiate the area formula with respect to radius.
- Chain volume rate into surface rate for a cube.
- Read marginal cost as the slope at a level.
Try at home. Find dA/dt for a circle with radius growing 2 cm/s at r = 6 cm.
Increasing and Decreasing Functions
Part of: Rates and Rising Graphs
A function climbs where its derivative stays positive and falls where it stays negative, provided it stays continuous across. Zeros of the derivative cut the line into test intervals, and a squared derivative that never turns negative keeps the function rising everywhere.
Why it matters. Monotonicity shapes graphs and gates maxima and minima.
Formulas
- Positive slope means rising; negative means falling
- Derivative zeros cut test intervals
- Squared slopes keep functions rising
Key words
- Increasing function
- : A function climbing across an interval.
- Decreasing function
- : A function sliding across an interval.
- Strictly monotonic
- : Always climbing or always sliding, never flat.
- Critical point
- : A point where the derivative is zero or missing.
Try it in two minutes
- Differentiate and find where the slope is zero.
- Test the sign in each cut interval.
- Name each interval rising or falling.
Try at home. Find rising and falling intervals of f(x) = x² − 2x − 3.
Local Maxima and Minima
Part of: Maxima, Minima and Best Values
Local peaks and valleys sit at critical points where the slope vanishes. The first derivative test watches the sign flip across, while the second derivative test reads the bend: upward bend means valley, downward means peak, and flat bend means the test fails.
Why it matters. Peaks and valleys locate best prices, fastest routes and safest designs.
Formulas
- Zero slope marks candidates, never verdicts
- Positive bend means valley; negative means peak
- Flat bend sends you back to sign test
Key words
- Local maximum
- : A peak value beating its near neighbours.
- Local minimum
- : A valley value beaten by its near neighbours.
- First derivative test
- : Reading the sign flip of slope across a point.
- Second derivative test
- : Reading the bend at a zero-slope point.
- Point of inflection
- : A flat crossing that is neither peak nor valley.
Try it in two minutes
- Differentiate and solve for zero-slope points.
- Test the sign on each side of every point.
- Confirm with the bend where it works.
Try at home. Find local extrema of f(x) = x³ − 3x + 1 with both tests.
Absolute Extrema and Best Choices
Part of: Maxima, Minima and Best Values
On a closed interval the champion values hide among critical points and the two ends. Word problems translate a fixed total into one variable, differentiate, and verify the bend or the ends before declaring the best choice.
Why it matters. Best-area, best-product and shortest-distance decisions all run this pipeline.
Formulas
- Compare critical points plus both ends
- Fixed sum splits evenly for max product
- Verify bend or ends before crowning
Key words
- Absolute maximum
- : The largest value on the whole given interval.
- Absolute minimum
- : The smallest value on the whole given interval.
- Optimisation
- : Choosing variables for the best possible value.
- Constraint
- : A fixed total linking the variables.
Try it in two minutes
- List critical points plus both ends.
- Evaluate the function at each candidate.
- Crown the largest and smallest values.
Try at home. Split 24 into two positives for maximum product.
What your child will practise
- Absolute Extrema and Best Choices: Desert Puzzle (Clue Puzzle)
- Absolute Extrema and Best Choices: Jungle Quick Fire (Quick Fire)
- Increasing and Decreasing Functions: Space Puzzle (Clue Puzzle)
- Local Maxima and Minima: Candy Quest (Flip and Match)
- Rates of Change: Space Quick Fire (Quick Fire)
- Quiz: Absolute Extrema and Best Choices (8 questions)
- Quiz: Increasing and Decreasing Functions (8 questions)
- Quiz: Local Maxima and Minima (8 questions)
- Quiz: Rates of Change (8 questions)