Math · Calculus

Applications of Derivatives formulas for JEE

Every Applications of Derivatives formula you need for JEE, grouped by concept.

FormulasRevision notes
26 formulas4 concepts
01Rate of Change02Increasing and Decreasing Functions03Tangents and Normals04Maxima and Minima
01

Rate of Change

3 formulas

Chain Rule for Related Rates

dydt=dydx⋅dxdt\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}

Calculates the rate of change of one variable with respect to time using its relation to another variable.

applies whendxdt≠0\frac{dx}{dt} \neq 0
rate of changederivatives

Marginal Cost

MC=dCdxMC = \frac{dC}{dx}

The instantaneous rate of change of total cost C(x) at any level of output x.

applies whenC(x) is a differentiable cost function.
economicsmarginal costrate of change

Marginal Revenue

MR=dRdxMR = \frac{dR}{dx}

The rate of change of total revenue R(x) with respect to the number of items sold x.

applies whenR(x) is a differentiable revenue function.
economicsmarginal revenuerate of change
02

Increasing and Decreasing Functions

2 formulas

Condition for Decreasing Function

f′(x)<0f'(x) < 0

A function f is strictly decreasing in an interval (a, b) if its first derivative is negative for each point in the interval.

applies whenf is continuous on [a, b] and differentiable on (a, b).
monotonicitydecreasing

Condition for Increasing Function

f′(x)>0f'(x) > 0

A function f is strictly increasing in an interval (a, b) if its first derivative is positive for each point in the interval.

applies whenf is continuous on [a, b] and differentiable on (a, b).
monotonicityincreasing
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03

Tangents and Normals

8 formulas

Angle of Intersection of Two Curves

tan⁡θ=∣m1−m21+m1m2∣\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|

The acute angle between the tangents to two intersecting curves at their point of intersection.

applies whenm1*m2 != -1 for non-orthogonal intersections.
jee-advancedangleintersection

Length of Normal

LN=∣y11+(dydx)2∣L_N = \left| y_1 \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \right|

The length of the normal segment intercepted between the point on the curve and the x-axis.

applies wheny1 != 0
jee-advancednormallength

Length of Subnormal

LSN=∣y1dydx∣L_{SN} = \left| y_1 \frac{dy}{dx} \right|

The length of the projection of the normal segment on the x-axis.

jee-advancedsubnormallength

Length of Subtangent

LST=∣y1dxdy∣L_{ST} = \left| y_1 \frac{dx}{dy} \right|

The length of the projection of the tangent segment on the x-axis.

applies whendy/dx != 0
jee-advancedsubtangentlength

Length of Tangent

LT=∣y11+(dxdy)2∣L_T = \left| y_1 \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \right|

The length of the tangent segment intercepted between the point of tangency and the x-axis.

applies wheny1 != 0
jee-advancedtangentlength

Equation of Normal Line

y−y1=−1f′(x1)(x−x1)y - y_1 = -\frac{1}{f'(x_1)}(x - x_1)

The equation of the normal to the curve y = f(x) at the point (x1, y1), orthogonal to the tangent.

applies whenFunction f(x) must be differentiable at x1 and f'(x1) != 0.
jee-advancednormalline

Shortest Distance Between Curves

dmin  ⟹  n^1∥n^2d_{min} \implies \hat{n}_1 \parallel \hat{n}_2

The shortest distance between two non-intersecting differentiable curves is always along their common normal.

applies whenCurves must be non-intersecting and differentiable.
jee-advancedshortest distancenormal

Equation of Tangent Line

y−y1=f′(x1)(x−x1)y - y_1 = f'(x_1)(x - x_1)

The point-slope form equation of the tangent to the curve y = f(x) at the point (x1, y1).

applies whenFunction f(x) must be differentiable at x1.
jee-advancedtangentline
04

Maxima and Minima

13 formulas

First Derivative Test for Maxima

f′(x) changes from + to −f'(x) \text{ changes from } + \text{ to } -

If f'(x) changes sign from positive to negative as x increases through c, then c is a point of local maxima.

applies whenc is a critical point where f is continuous.
optimizationmaximafdt

Second Derivative Test for Maxima

f′(c)=0 and f′′(c)<0f'(c) = 0 \text{ and } f''(c) < 0

A critical point c is a point of local maxima if the first derivative is zero and the second derivative is negative.

applies whenf must be twice differentiable at c.
optimizationmaximasdt

First Derivative Test for Minima

f′(x) changes from − to +f'(x) \text{ changes from } - \text{ to } +

If f'(x) changes sign from negative to positive as x increases through c, then c is a point of local minima.

applies whenc is a critical point where f is continuous.
optimizationminimafdt

Second Derivative Test for Minima

f′(c)=0 and f′′(c)>0f'(c) = 0 \text{ and } f''(c) > 0

A critical point c is a point of local minima if the first derivative is zero and the second derivative is positive.

applies whenf must be twice differentiable at c.
optimizationminimasdt

n-th Derivative Test

f(n)(c)≠0 and f(k)(c)=0 for k<nf^{(n)}(c) \neq 0 \text{ and } f^{(k)}(c) = 0 \text{ for } k < n

If the first non-zero derivative at a critical point is of even order, it is an extremum (max if < 0, min if > 0). If it is of odd order, it's a point of inflection.

applies whenRequires existence of continuous higher-order derivatives.
jee-advancedoptimizationhigher order test

Least CSA Cone for Given Volume

h=2rh = \sqrt{2} r

The altitude of a right circular cone of least curved surface area and a given volume.

applies whenCone volume is constant.
optimizationgeometrycone

Max Volume Cone given Surface Area

α=sin⁡−1(13)\alpha = \sin^{-1}\left(\frac{1}{3}\right)

The semi-vertical angle of the right circular cone of given surface area and maximum volume.

applies whenTotal surface area is constant.
optimizationgeometryconeangle

Max Volume Cone given Slant Height

α=tan⁡−1(2)\alpha = \tan^{-1}(\sqrt{2})

The semi-vertical angle of the cone of maximum volume for a given fixed slant height.

applies whenSlant height is constant.
optimizationgeometryconeangle

Altitude of Max Volume Cone in Sphere

h=43Rh = \frac{4}{3} R

The altitude of the largest right circular cone that can be inscribed in a sphere of radius R.

applies whenSphere radius R is fixed.
optimizationgeometryconesphere

Max Volume of Cone Inscribed in Sphere

Vcone=827VsphereV_{cone} = \frac{8}{27} V_{sphere}

The volume of the largest right circular cone that can be inscribed in a sphere.

applies whenSphere radius R is fixed.
optimizationgeometryconesphere

Height of Max Volume Cylinder in Cone

hcyl=13hconeh_{cyl} = \frac{1}{3} h_{cone}

The height of the right circular cylinder of greatest volume inscribed in a right circular cone.

applies whenCone dimensions are fixed.
optimizationgeometrycylindercone

Height of Max Volume Cylinder in Sphere

h=2R3h = \frac{2R}{\sqrt{3}}

The height of the cylinder of maximum volume that can be inscribed in a sphere of radius R.

applies whenSphere radius R is fixed.
optimizationgeometrycylindersphere

Radius of Curvature

R=(1+(y′)2)3/2∣y′′∣R = \frac{(1 + (y')^2)^{3/2}}{|y''|}

The radius of the osculating circle representing the local curvature of a curve.

applies wheny'' != 0
jee-advancedcurvature
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