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JEE Mains Maths · Formula sheet

Application of Derivatives formulas

17 formulas and 17 common traps for JEE Mains Maths Application of Derivatives, grouped by subtopic.

Full notes with worked examples

Tangents, Normals and Rates of Change

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Rates of change

Chain rule for rates

dVdt=dVdr⋅drdt\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}

Tangents and normals

Tangent at a point

y−y0=f′(x0)(x−x0)y-y_0=f'(x_0)(x-x_0)

Common traps

Differentiate before substituting

Put the instant's values in only after differentiating. Substituting h=10h=10 first turns a variable into a constant and its rate into 0.

A normal parallel to a line

If the normal is parallel to a line of slope mm, the tangent has slope −1m-\frac1m. Setting f′(x)=mf'(x)=m finds points where the tangent, not the normal, is parallel to the line.

Increasing and Decreasing Functions

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The sign of the derivative

Monotonicity test

f′(x)>0 on I ⇒ f increasing on If'(x)>0\ \text{on }I\ \Rightarrow\ f\ \text{increasing on }I

Parameters that make a function monotonic

Quadratic never negative

ax2+bx+c≥0 ∀x ⇐ a>0, b2−4ac≤0ax^2+bx+c\ge0\ \forall x\ \Leftarrow\ a>0,\ b^2-4ac\le0

Increasing derivatives and order

Symmetric sum

g(x)=f(x)+f(c−x): g′(x)=f′(x)−f′(c−x)g(x)=f(x)+f(c-x):\ g'(x)=f'(x)-f'(c-x)

Common traps

Each piece is not the union

1x\frac1x decreases on (−∞,0)(-\infty,0) and on (0,∞)(0,\infty), but 1−1<11\frac1{-1}<\frac11, so it does not decrease on their union. Options often hinge on this.

Include the boundary value

At the boundary value f′f' touches 0 at one point, and the function still increases. So 'increasing for all x' usually gives a closed condition like λ≤13\lambda\le\frac13.

f″ > 0 is about f′, not f

f′′>0f''>0 makes f′f' increase; ff itself can still fall wherever f′<0f'<0. Keep track of which function the sign statement is about.

Counting Real Roots

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A monotonic function has at most one root

At most one root

f′(x)>0 ∀x ⇒ f(x)=0 has at most one rootf'(x)>0\ \forall x\ \Rightarrow\ f(x)=0\ \text{has at most one root}

Counting roots from the extreme values

Three roots of a cubic

f(x)=k has 3 distinct roots ⇐ fmin⁡<k<fmax⁡f(x)=k\ \text{has 3 distinct roots}\ \Leftarrow\ f_{\min}<k<f_{\max}

Common traps

At most one is not exactly one

Monotonicity gives at most one root. It is exactly one only when ff also takes both signs — check two values, or the limits at the ends.

Count the ends too

The first and last crossings come from the limits at ±∞\pm\infty. An even-degree polynomial with positive leading term starts and ends positive, so a single minimum below 0 gives two roots, not one.

Local Maxima and Minima

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Sign changes of the derivative

Derivative of an integral

ddx∫ag(x)h(t) dt=h(g(x)) g′(x)\frac{d}{dx}\int_a^{g(x)}h(t)\,dt=h\big(g(x)\big)\,g'(x)

Corners, cusps and piecewise functions

Critical points

f′(c)=0 or f′(c) does not exist (c∈domain)f'(c)=0\ \text{or}\ f'(c)\ \text{does not exist}\ (c\in\text{domain})

Common traps

Even powers do not change sign

A zero of f′f' from a factor like (t−4)6(t-4)^6 is a critical point but not an extremum. Count only the zeros where the sign actually flips.

A corner need not be an extremum

If the function falls on both sides of a corner, the corner is not an extremum. Check the direction on each side instead of counting every kink.

Functions Built from Their Extrema

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A polynomial from its extrema

Start from the derivative

f′(x)=k(x−a)(x−b) ⇒ f(x)=k(x33−a+b2x2+abx)+Cf'(x)=k(x-a)(x-b)\ \Rightarrow\ f(x)=k\left(\tfrac{x^3}3-\tfrac{a+b}2x^2+abx\right)+C

Where a cubic's maximum and minimum sit

Order of the extrema

f(x)=ax3+…, a>0: xmax⁡<xmin⁡f(x)=ax^3+\dots,\ a>0:\ x_{\max}<x_{\min}

Common traps

Use every condition once

Count unknowns and conditions: each extremum gives one equation, the limit fixes several coefficients at once, and a value gives one more. A condition used twice leaves a constant undetermined.

Check the sign of the leading term

With a negative leading coefficient, or a parameter that may be negative, the smaller root of f′f' is the minimum. Settle the sign before naming which root is which.

Greatest and Least Values

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Closed interval: critical points and endpoints

Closed-interval method

max⁡[a,b]f=max⁡{f(a), f(b), f(c):f′(c)=0}\max_{[a,b]}f=\max\{f(a),\,f(b),\,f(c):f'(c)=0\}

Ranges and least values over a whole domain

AM-GM

a+b≥2ab(a,b>0), equality when a=ba+b\ge2\sqrt{ab}\quad(a,b>0),\ \text{equality when }a=b

Common traps

The endpoints count

A local maximum inside the interval can be smaller than the value at an endpoint. In (x+3)2(x−2)3(x+3)^2(x-2)^3 on [−4,4][-4,4], both the greatest and the least values sit at endpoints.

Is the bound reached?

AM-GM and the discriminant give bounds. Check that the equality case happens at an allowed xx; if it does not, the bound is not the least value.

Optimisation Problems

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Largest area or volume

The method

Q=Q(x),Q′(x)=0,Q′′(x)<0 ⇒ maximumQ=Q(x),\quad Q'(x)=0,\quad Q''(x)<0\ \Rightarrow\ \text{maximum}

Shortest distances

Nearest point to a line

f′(x0)=slope of the linef'(x_0)=\text{slope of the line}

Common traps

Stay inside the allowed range

The cut in a box must be less than half the sheet's side, and a wire piece cannot be negative. A root of Q′Q' outside that range is not the answer.

Only if they do not meet

The parallel-tangent method assumes the curve and the line do not cross; if they meet, the shortest distance is 0. Check with a discriminant first.

Rolle's and Mean Value Theorems

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Rolle's theorem and the mean value theorem

Mean value theorem

f′(c)=f(b)−f(a)b−a,c∈(a,b)f'(c)=\frac{f(b)-f(a)}{b-a},\quad c\in(a,b)

Zeros of f′ and f″

Rolle, repeated

n zeros of f ⇒ ≥n−1 zeros of f′n\ \text{zeros of }f\ \Rightarrow\ \ge n-1\ \text{zeros of }f'

Common traps

Check the hypotheses

Rolle's theorem needs differentiability inside the interval. ∣x∣|x| on [−1,1][-1,1] has equal end values but no point with f′=0f'=0.

The zeros must be distinct

Rolle needs two different points with equal values. A repeated root counts once for this argument, so count the distinct zeros before subtracting one.

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