JEE Mains Maths · Application of Derivatives

Tangents, Normals and Rates of Change

The derivative as a slope and as a rate: tangents and normals to a curve, curves meeting at an angle, and quantities that change together over time.

All Application of Derivatives notes JEE Mains Maths strategy

Why this matters

Twenty PYQs, eleven of them multiple choice. Four are rates of change — a balloon, a cone filling with water. Sixteen use the slope of a curve: tangents and normals at a point, tangents through an outside point, and curves that cut at right angles. Two ideas cover the page.

Concept 1 of 2: Rates of change

When two quantities are tied by a formula, their rates are tied by its derivative. Write the formula in one variable first (use similar triangles for a cone), differentiate with respect to time by the chain rule, and only then put in the numbers for the instant asked about.

Definition

$\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$ .
Sphere: $V=\frac43\pi r^3$ , $S=4\pi r^2$ ; $\frac{dV}{dt}=S\frac{dr}{dt}$ .
Cone: $V=\frac13\pi r^2h$ , curved surface $\pi rl$ ; a fixed cone fixes $\frac rh$ .
A quantity growing at a constant rate is linear in time.

Chain rule for rates

\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}

Worked example

The radius of a sphere grows at

0.1

cm/s. How fast is its volume growing when

r=5

cm?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 14 June 2022 · Q63Moderate

Example 1 · Application of Derivatives · Tangents, Normals and Rates of Change

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is:

Differentiate before substituting

Put the instant's values in only after differentiating. Substituting

h=10

first turns a variable into a constant and its rate into 0.

Drill 3 more on rates of change

Concept 2 of 2: Tangents and normals

The slope of the tangent at

x_0

f'(x_0)

, and the normal is at right angles to it, with slope

-\frac1{f'(x_0)}

. For a curve given by a parameter,

\frac{dy}{dx}=\frac{dy/dt}{dx/dt}

; for an implicit curve, differentiate both sides. A tangent 'through an outside point' is found by writing the tangent at a general point and making it pass through that point.

Definition

Tangent: $y-y_0=m(x-x_0)$ , $m=f'(x_0)$ ; normal slope $-\frac1m$ .
Horizontal tangent: $\frac{dy}{dx}=0$ ; vertical: $\frac{dx}{dy}=0$ .
Parametric: $\frac{dy}{dx}=\frac{\dot y}{\dot x}$ .
Angle between curves: $\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|$ ; at right angles, $m_1m_2=-1$ .

Tangent at a point

y-y_0=f'(x_0)(x-x_0)

Worked example

Find the tangent and the normal to

y=x^2-3x

x=1

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 30 January 2023 · Q68Moderate

Example 2 · Application of Derivatives · Tangents, Normals and Rates of Change

The number of points on the curve

y = 54x^{5}- 135x^{4}- 70x^{3}+ 180x^{2}+ 210x

at which the normal lines are parallel to

x + 90y + 2 = 0

is :

A normal parallel to a line

If the normal is parallel to a line of slope

m

, the tangent has slope

-\frac1m

. Setting

f'(x)=m

finds points where the tangent, not the normal, is parallel to the line.

Drill 15 more on tangents and normals

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Rates of change
Chain rule for rates
$\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}$
Tangents and normals
Tangent at a point
$y-y_0=f'(x_0)(x-x_0)$

Watch out for (2)

Differentiate before substituting
→ Rates of change
A normal parallel to a line
→ Tangents and normals

Test yourself on Application of Derivatives

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Application of Derivatives chapter test Drill all 20 questions