JEE Mains Maths · Differentiation

Implicit, Parametric and Logarithmic Differentiation

Finding dy/dx and d²y/dx² when y is tied to x by an equation or a parameter, or is a power or product that is easier after taking logarithms.

All Differentiation notes JEE Mains Maths strategy

Why this matters

Ten PYQs, six of them multiple choice. Five differentiate an equation in x and y, or a curve given by a parameter, four of them twice; five use logarithms — two powers like xˣ, a long product, a logarithm of a quotient, and ln y given as a function of x. Two ideas cover the page.

Concept 1 of 2: Implicit and parametric differentiation

When

x

and

y

are tied by one equation, differentiate every term with respect to

x

and treat

y

as a function:

y^2

gives

2yy'

. Put in the point before solving for

y'

; the arithmetic is much lighter. For a curve given by a parameter

t

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

, and the second derivative needs one more division by

\frac{dx}{dt}

Definition

Differentiate both sides; each $y$ term picks up a factor $y'$ .
Put in the point, then solve for $y'$ ; differentiate again for $y''$ .
Parametric: $\frac{dy}{dx}=\frac{\dot y}{\dot x}$ and $\frac{d^2y}{dx^2}=\frac{\dot x\ddot y-\dot y\ddot x}{\dot x^3}$ .

Second derivative, parametric

\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\Big/\frac{dx}{dt}

Worked example

Find

y'

and

y''

(1,2)

on the curve

x^2+xy+y^2=7

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 6 April 2023 · Q77Moderate

Example 1 · Differentiation · Implicit, Parametric and Logarithmic Differentiation

2x^{y}+ 3y^{x}= 20

, then

\frac{dy}{dx}

(2,2)

is equal to

Not the ratio of second derivatives

For a parametric curve

\frac{d^2y}{dx^2}\ne\frac{\ddot y}{\ddot x}

. With

x=t^2,\ y=t^3

the ratio gives

3t

, but the true value is

\frac{3}{4t}

. Differentiate

\frac{dy}{dx}

with respect to

t

, then divide by

\frac{dx}{dt}

Drill 4 more on implicit and parametric differentiation

Concept 2 of 2: Logarithmic differentiation

A variable power like

x^x

has no ordinary rule. Take logarithms:

\ln y=x\ln x

, so

\frac{y'}{y}=\ln x+1

. The same step turns a long product into a sum and a quotient into a difference. When

\ln y

is given, as in

\ln y=k\sin^{-1}x

, clear the square root and differentiate again: the question usually wants a combination of

y''

y'

and

y

, not each one.

Definition

$y=f(x)^{g(x)}$ : $\ln y=g\ln f$ , so $\frac{y'}{y}=g'\ln f+\frac{gf'}{f}$ .
A product becomes a sum of logarithms; a quotient becomes a difference.
$\frac{dx}{dy}=\frac{1}{y'}$ and $\frac{d^2x}{dy^2}=-\frac{y''}{(y')^3}$ .

Variable power

\frac{d}{dx}f^{\,g}=f^{\,g}\left(g'\ln f+\frac{g\,f'}{f}\right)

Worked example

Find

y'(e)

y=x^{\ln x}

x>0

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 29 January 2024 · Q167Moderate

Example 2 · Differentiation · Implicit, Parametric and Logarithmic Differentiation

Let

y =\log_{e}\left( \frac{1 -x^{2}}{1 +x^{2}} \right), - 1 < x < 1

. Then at

x =\frac{1}{2}

, the value of

225\left( y^{'}-y^{''} \right)

is equal to

d²x/dy² is not 1/y″

\frac{dx}{dy}=\frac{1}{y'}

, but

\frac{d^2x}{dy^2}=-\frac{y''}{(y')^3}

, not

\frac{1}{y''}

. Differentiate

\frac{1}{y'}

with respect to

x

, then divide by

y'

once more.

Drill 4 more on logarithmic differentiation

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)

Implicit and parametric differentiation
Second derivative, parametric
$\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\Big/\frac{dx}{dt}$
Logarithmic differentiation
Variable power
$\frac{d}{dx}f^{\,g}=f^{\,g}\left(g'\ln f+\frac{g\,f'}{f}\right)$

Watch out for (2)

Not the ratio of second derivatives
→ Implicit and parametric differentiation
d²x/dy² is not 1/y″
→ Logarithmic differentiation

Test yourself on Differentiation

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.

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