# JEE Main

Explore popular questions from Differential Calculus for JEE Main. This collection covers Differential Calculus previous year JEE Main questions hand picked by experienced teachers.

## Mathematics

Differential Calculus

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Q 1. If {tex} y = \sec \left( \tan ^ { - 1 } x \right) , {/tex} then {tex} \frac { d y } { d x } {/tex} at {tex} x = 1 {/tex} is equal to

A

{tex} \frac { 1 } { 2 } {/tex}

B

1

C

{tex} \sqrt { 2 } {/tex}

{tex} \frac { 1 } { \sqrt { 2 } } {/tex}

##### Explanation

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Q 2. If {tex} y = e ^ { n x } , {/tex} then {tex} \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d ^ { 2 } x } { d y ^ { 2 } } \right) {/tex} is equal to

A

{tex} n e ^ { n x } {/tex}

B

{tex} n e ^ { - n x } {/tex}

C

1

{tex} - n e ^ { - n x } {/tex}

##### Explanation

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Q 3. If {tex} \cos ( x + y ) = y \sin x , {/tex} then {tex} \frac { d y } { d x } {/tex} is equal to

{tex} - \frac { \sin ( x + y ) + y \cos x } { \sin x + \sin ( x + y ) } {/tex}

B

{tex} \frac { \sin ( x + y ) + y \cos x } { \sin x + \sin ( x + y ) } {/tex}

C

{tex} \frac { y \cos x - \sin ( x + y ) } { \sin x - \sin ( x + y ) } {/tex}

D

None of these

##### Explanation

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Q 4. If {tex} \sin ^ { 2 } x + 2 \cos y + x y = 0 , {/tex} then {tex} \frac { d y } { d x } {/tex} is equal to

A

{tex} \frac { y + 2 \sin x } { 2 \sin y + x } {/tex}

{tex} \frac { y + \sin 2 x } { 2 \sin y - x } {/tex}

C

{tex} \frac { y + 2 \sin x } { \sin y + x } {/tex}

D

None of these

##### Explanation

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Q 5. If {tex} x ^ { 3 } + 8 x y + y ^ { 3 } = 64 , {/tex} then {tex} \frac { d y } { d x } {/tex} is equal to

{tex} - \frac { 3 x ^ { 2 } + 8 y } { 8 x + 3 y ^ { 2 } } {/tex}

B

{tex} \frac { 3 x ^ { 2 } + 8 y } { 8 x + 3 y ^ { 2 } } {/tex}

C

{tex} \frac { 3 x + 8 y ^ { 2 } } { 8 x ^ { 2 } + 3 y } {/tex}

D

None of these

##### Explanation

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Q 6. If {tex} \sin ( x + y ) = \log ( x + y ) , {/tex} then {tex} \frac { d y } { d x } {/tex} is equal to

A

{tex}2{/tex}

B

{tex} - 2 {/tex}

C

{tex}1{/tex}

{tex} - 1 {/tex}

##### Explanation

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Q 7. If In {tex} ( x + y ) = 2 x y , {/tex} then {tex} y ^ { \prime } ( 0 ) = {/tex}

1

B

{tex} - 1 {/tex}

C

2

D

0

##### Explanation

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Q 8. If {tex} x ^ { y } = e ^ { x - y } , {/tex} then {tex} \frac { d y } { d x } {/tex} is equal to

{tex} \log x \cdot [ \log ( e x ) ] ^ { - 2 } {/tex}

B

{tex} \log x \cdot [ \log ( e x ) ] ^ { 2 } {/tex}

C

{tex} \log x \cdot ( \log x ) ^ { 2 } {/tex}

D

None of these

##### Explanation

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Q 9. If {tex} y = ( \tan x ) ^ { ( \tan x ) ^ { \tan x } } , {/tex} then at {tex} x = \frac { \pi } { 4 } , {/tex} the value of {tex} \frac { d y } { d x } {/tex} is equal to

A

0

B

1

2

D

None of these

##### Explanation

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Q 10. If {tex} y = ( \sin x ) ^ { \tan x } , {/tex} then {tex} \frac { d y } { d x } {/tex} is equal to

{tex} ( \sin x ) ^ { \tan x } \cdot \left( 1 + \sec ^ { 2 } x \cdot \log \sin x \right) {/tex}

B

{tex} \tan x \cdot ( \sin x ) ^ { \tan x - 1 } \cdot \cos x {/tex}

C

{tex} ( \sin x ) ^ { \tan x } \cdot \sec ^ { 2 } x \cdot \log \sin x {/tex}

D

{tex} \tan x \cdot ( \sin x ) ^ { \tan x - 1 } {/tex}

##### Explanation

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Q 11. If {tex} y = 2 ^ { 1 / \log _ { x } 4 } , {/tex} then {tex} x {/tex} is equal to

A

{tex} \sqrt { y } {/tex}

B

{tex} y {/tex}

{tex} y ^ { 2 } {/tex}

D

{tex} y ^ { 4 } {/tex}

##### Explanation

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Q 12. The derivative of {tex} y = x ^ { \ln x } {/tex} is

A

{tex} x ^ { \ln x } \ln x {/tex}

B

{tex} x ^ { \ln x - 1 } \ln x {/tex}

2{tex} x ^ { \ln x - 1 } \ln x {/tex}

D

{tex} x ^ { \ln x - 2 } {/tex}

##### Explanation

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Q 13. If {tex} y = \sin \left( 2 \sin ^ { - 1 } x \right) , {/tex} then {tex} \frac { d y } { d x } {/tex} is equal to

{tex} \frac { 2 - 4 x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } {/tex}

B

{tex} \frac { 2 + 4 x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } {/tex}

C

{tex} \frac { 2 - 4 x ^ { 2 } } { \sqrt { 1 + x ^ { 2 } } } {/tex}

D

{tex} \frac { 2 + 4 x ^ { 2 } } { \sqrt { 1 + x ^ { 2 } } } {/tex}

##### Explanation

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Q 14. If {tex} y = \cos ^ { - 1 } \left( \frac { 3 \cos x + 4 \sin x } { 5 } \right) , {/tex} then {tex} \frac { d y } { d x } {/tex}

A

0

1

C

- 1

D

{tex} \frac { 1 } { 2 } {/tex}

##### Explanation

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Q 15. {tex} \frac { d } { d x } \cos ^ { - 1 } \frac { x - x ^ { - 1 } } { x + x ^ { - 1 } } {/tex} is equal to

A

{tex} \frac { 1 } { 1 + x ^ { 2 } } {/tex}

B

{tex} \frac { - 1 } { 1 + x ^ { 2 } } {/tex}

C

{tex} \frac { 2 } { 1 + x ^ { 2 } } {/tex}

{tex} \frac { - 2 } { 1 + x ^ { 2 } } {/tex}

##### Explanation

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Q 16. {tex} \frac { d } { d x } \cos ^ { - 1 } \sqrt { \frac { 1 + x ^ { 2 } } { 2 } } {/tex} is equal to

A

{tex} \frac { - 1 } { 2 \sqrt { 1 - x ^ { 4 } } } {/tex}

B

{tex} \frac { 1 } { 2 \sqrt { 1 - x ^ { 4 } } } {/tex}

{tex} \frac { - x } { \sqrt { 1 - x ^ { 4 } } } {/tex}

D

{tex} \frac { x } { \sqrt { 1 - x ^ { 4 } } } {/tex}

##### Explanation

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Q 17. Differential coefficient of {tex} \sin ^ { - 1 } \frac { 1 - x } { 1 + x } {/tex} with respect to {tex} \sqrt { x } {/tex} is

A

{tex} \frac { 1 } { 2 \sqrt { x } } {/tex}

B

{tex} \frac { \sqrt { x } } { \sqrt { 1 - x } } {/tex}

C

1

None of these

##### Explanation

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Q 18. The differential coefficient of {tex} \tan ^ { - 1 } \frac { 2 x } { 1 - x ^ { 2 } } {/tex} with respect to {tex} \sin ^ { - 1 } \frac { 2 x } { 1 + x ^ { 2 } } {/tex} is

1

B

{tex} - 1 {/tex}

C

0

D

None of these

##### Explanation

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Q 19. If {tex} y = x \log \left( \frac { x } { a + b x } \right) , {/tex} then {tex} x ^ { 3 } \frac { d ^ { 2 } y } { d x ^ { 2 } } {/tex} is equal to

A

{tex} x \frac { d y } { d x } - y {/tex}

{tex} \left( x \frac { d y } { d x } - y \right) ^ { 2 } {/tex}

C

{tex} y \frac { d y } { d x } - x {/tex}

D

{tex} \left( y \frac { d y } { d x } - x \right) ^ { 2 } {/tex}

##### Explanation

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Q 20. If {tex} y = x ^ { 3 } \log \log _ { e } ( 1 + x ) , {/tex} then {tex} y ^ { \prime \prime } ( 0 ) {/tex} equals

0

B

- 1

C

6 {tex} \log _ { e } 2 {/tex}

D

6

##### Explanation

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Q 21. {tex} \frac { d ^ { 2 } x } { d y ^ { 2 } } {/tex} is equal to

A

{tex} \frac { 1 } { ( d y / d x ) ^ { 2 } } {/tex}

B

{tex} \frac { \left( d ^ { 2 } y / d x ^ { 2 } \right) } { ( d y / d x ) ^ { 2 } } {/tex}

C

{tex} \frac { d ^ { 2 } y } { d x ^ { 2 } } {/tex}

{tex} \frac { \left( - d ^ { 2 } y / d x ^ { 2 } \right) } { ( d y / d x ) ^ { 2 } } {/tex}

##### Explanation

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Q 22. For all {tex} x \in ( 0,1 ) , {/tex} which is the correct one?

A

{tex} e ^ { x } < 1 + x {/tex}

{tex} \log _ { e } ( 1 + x ) < x {/tex}

C

{tex} \sin x > x {/tex}

D

{tex} \log _ { e } x > x {/tex}

##### Explanation

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Q 23. The function {tex} f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + 90 x + 174 {/tex} is increasing in the interval

A

{tex} \frac { 1 } { 2 } < x < 1 {/tex}

B

{tex} \frac { 1 } { 2 } < x < 2 {/tex}

C

{tex} 3 < x < \frac { 59 } { 4 } {/tex}

{tex} - \infty < x < \infty {/tex}

##### Explanation

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Q 24. The function {tex} f ( x ) = \left( \frac { e ^ { 2 x } - 1 } { e ^ { 2 x } + 1 } \right) {/tex} is

Increasing

B

Decreasing

C

Even

D

None of these

##### Explanation

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Q 25. What is the value of {tex} x {/tex} if the function {tex} x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } - 10 {/tex} has a maximum?

A

3

B

2

1

D

0