Practice Question on Differentiation Solved and Unsolved (Hard)

Differentiation is a fundamental concept in calculus that measures how a function changes as input changes. It is used in various fields, such as physics, engineering, and economics, to find rates of change, the Slope of curves, and optimization solutions.

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Important Formulas of Differentiation

Below are some basic differentiation formulas to help you solve the questions:

Check - Differentiation Fomula

Solved Practice Problems on Differentiation (Hard)

The following question focuses on differentiation at a Hard level.

Question 1: Find dy /dx if x³ + y³ = 6xy

Solution:

x³ + y³ = 6xy

Differentiate both sides w.r.t. x:

3x² + 3y² dy/dx = 6y + 6x dy/dx
3y² dy/dx - 6x dy/dx = 6y - 3x²
dy /dx ( 3y² - 6x) = 6y - 3x²
dy/dx = 6y - 3x² / 3y² - 6x

Question 2: Find dy/dx if y = x^x / e^x

Solution :

y = x^x / e^x
Rewrite using logarithm
y = x^xe^-x
take log on both sides
ln y = x ln x - x
Differentiating with respect to x, we get;
1/y .dy/dx = lnx +1 - 1
dy/dx = y(ln x) = x^x/ e^x (ln x)
dy/dx = x^xe^-x ln x

Question 3: If y = (x + 1)^sinx find dy/dx.

Solution:

Take log at both sides
ln y = sinx ln(x + 1)
Differentiating with respect to x, we get;
1/y dy/dx = cosx ln(x + 1) + sinx/ x + 1
dy/dx = (x + 1)^sinx(cosx ln (x + 1) + sinx /x + 1

Question 4: If x = y^y then find dy/dx

Solution:

x = y^y
taking log to both sides
ln x = y^y
1/x dy/dx =ln x + 1 +y.1/y.dy/dx
1/x = ln y + 1 + dy/dx
dy/dx = 1/x - (ln y + 1)
substituing x = y ^y
dy/dx = 1/y^y - (ln y + 1)

Question 5: If log_e (x + y) = 4xy, find (d²y)/(dx²) at x = 0.

Solution:

Given that loge (x + y) = 4xy

Differentiating with respect to x, we get
(1/(x + y)) [1 + (dy/dx)] = 4[x (dy/dx) + y]
1 + (dy/dx) = 4(x + y) [x (dy/dx) + y]⋯(i)
If x = 0, then y = 1.
From (i), we get
1 + dy/dx = 4
dy/dx = 3
Again differentiate (i) w.r.t. x.
d²y/dx² = 4(x + y)[x (d²y)/(dx²) + 2 (dy/dx)] + 4[x (dy/dx) + y](1 + (dy/dx))
At x = 0, y = 1, dy/dx = 3
d²y/dx² = 4(0 + 1)[0 + 2x3]+4[0 + 1](1 + 3)
= 40
So, d²y/dx² = 40.

Question 6: If √(1 – x²) + √(1 – y²) = a(x – y), then find dy/dx.

Solution:

Given that √(1 – x2) + √(1 – y2) = a(x – y) …(i)
Let x = sin A and y = sin B
A = sin^-1 x
B = sin^-1 y
So , from (i)
√(1 – sin² A) + √(1 – sin² B) = a(sin A – sin B)
cos A + cos B = a(sin A – sin B) ..…(ii) [1 - sin² θ = cosθ ]
We have cos A + cos B = 2 cos (A + B)/2 cos (A – B)/2
Also, sin A – sin B = 2 cos (A + B)/2 sin (A – B)/2
Substitute the above 2 equations in (ii)
2 cos (A + B)/2 cos (A – B)/2 = a 2 cos (A + B)/2 sin (A – B)/2
cos (A – B)/2 = a sin (A – B)/2
cot (A – B)/2 = a
(A – B) = 2 cot^-1 a
Substitute the values of A and B.
sin^-1 x – sin^-1 y = 2 cot^-1 a
Differentiating with respect to x, we get;
1/√(1 – x2) – 1/√(1 – y2) dy/dx = 0
1/√(1 – x2) = 1/√(1 – y2) dy/dx
dy/dx = √(1 – y2)/√(1 – x2)

Question 7: If log_e y = 3sin^-1 x,then find the value of (1 - x²)y^'' - xy^' at x=1/2 [ JEE Main-2024].

Solution:

log_ey = 3sin^-1x
y = e^{3sin^{-1}{x}}
dy /dx = e^{3sin^{-1}{x}} . 3 / √ 1 - x²
√ 1 - x² dy /dx = 3y
Again differentiate
√ 1 - x² . y" - 2x / 2√ 1 - x² y' = 3y^'
(1 - x ²)y^" - xy' = 3y^' (√ 1 - x²)
So the value of 3y^' (√ 1- x²) at x = 1/2
3.3/ √ 1 - x² .e^{sin^{-1}{x}}(√ 1 - x²)
9e^{3 π/6} = 9e ^c/2

Question 8: If f(x) = { x³sin (1/x) , x ≠ 0 0 , x=0 } find the value of f" [JEE Mains-2024].

Solution:

f(x) = { x³sin (1/x) , x ≠ 0 0 , x=0 }

We need to find second derivative at specific points
Firstly let's us compute the first derivative f'(x)
f'(x) = 3 x² (1/x) - x cos (1/x)
Next,the second order derivative f^"(x) is :
f"(x) = 6x sin (1/x) - 3x cos (1/x) - cos(1/x) - 1/x sin (1/x)

Therefore evaluting the second order derivative at x = 2/ π
f"(2/π) = 6(2/π) sin (π/2) - 3(2/π) cos (π/2) - cos(π/2) - π/2 sin (π/2)
Since sin(π/2) = 1 and cos (π/2) = 0 simplfies to:
f"(2/π) = 12/π - π/2 = 24 - π²/ 2π
Finally note that f'(0) is not defined ,as it involve terms like 1/x when x = 0.

Check- Differentiation Quiz

Unsolved Practice Question on Differentiation (Hard)

Question 1: Find dy /dx if x³ + y³ - 3xy = 0.

Question 2: Find the dy/dx if y= (x² + 1)⁵ / (x³ - 3x + 7)⁴.

Question 3: Find d²y/dx² if x² + y² = tan(y).

Question 4: If sin(y) + y^x =x^y + cosx find dy/dx.

Question 5: If sin^-1(x/√ 1 + x²) find dy/dx.

Question 6: If x^y + y^x = e^xy then find d²y/dx².

Question 7: If f(x) = { xⁿ sin (1/x) , x ≠ 0 0 , x=0 } find the value of f".

Question 8: If log_e y = 4sin^-1 x,then find the value of (1 - x²)y'' - xy' at x=1/2 .

Answer Sheet

1) dy/dx = y - x² / y² - x
2) dy/dx = (x² + 1 ) ⁵/(x³ - 3x + 7)⁴ (10x /x² + 1 -12x² -12 /x³ - 3x +7 )
3) d²y/dx^{2 =}= (2y- sec²y)(-2) -(-2x)(2dy/dx - sec²ytany dy/dx) / (2y - sec²y)²
4) dy/dx = -sinx /cosy
5) 1/√ 1 + x²
6) (2x−e^xyx)d/dx(e^xyy-2y) - (e^xyy - 2xy)d/dx(2x -e^xyx) / (2x - e^xyx)²
7) f"(0) exist if n > 4.
8) e^2π/3 (16 - 8√3 /3 )

Practice More:

• Practice Question on Differentiation (Easy)
• Practice Question on Differentiation (Medium)