## Differentiation Quiz

Important topics for Maths has been designed in such a way that it offers very practical and application-based learning to further make it easier for students to understand every concept or topic by correlating it with the day-to-day experiences.

Q1. If e^x=(√(1+t)-√(1-t))/(√(1+t)+√(1-t)) and tan⁡y/2=√((1-t)/(1+t),) then dy/dx at t=1/2 is
•  -1/2
•  1/2
•  0
•  None of these
Solution

Q2.d/dx √((1-sin⁡2x)/(1+sin⁡2x )) is equal to (0
•  sec^2⁡x
•  -sec^2⁡(Ï€/4-x)
•  sec^2⁡(Ï€/4+x)
•  sec^2⁡(Ï€/4-x)
Solution
(b) y=√((1-sin⁡2x)/(1+sin⁡2x ))=cos⁡x-sin⁡x /cos⁡x+sin⁡x =(1-tan⁡x)/(1+tan⁡x )=tan⁡(Ï€/4-x) ⇒ dy/dx=-sec^2⁡(Ï€/4-x)

Q3.

•   e
•  2e
•  4e
•  None of these
Solution

Q4. The differential coefficient of f(log_e ⁡x) with respect to x, where f(x)=log_e⁡x is
•  x/log_e ⁡x
•  1/x log_e ⁡x
•  1/(x log_e x)
•  None of these
Solution

Q5.If y√(x^2+1)=log⁡(√(x^2+1)-x), then (x^2+1) dy/dx+xy+1=
•  0
•  1
•  2
•  None of these
Solution
(a) y√(x^2+1)=log⁡{√(x^2+1)-x} Differentiating both sides w.r.t. x., we get dy/dx √(x^2+1)+y 1/(2√(x^2+1)) 2x=1/(√(x^2+1)-x)×{1/2 2x/√(x^2+1)-1} ⇒(x^2+1) dy/dx+xy=√(x^2+1) (-1)/√(x^2+1) ⇒(x^2+1) dy/dx+xy+1=0

Q6. dy/dx for y=tan^(-1)⁡{√((1+cos⁡x)/(1-cos⁡x ))}, where 0
•  -1/2
•  0
• 1
•  -1
Solution
(a) y=tan^(-1)⁡{√((1+cos⁡x)/(1-cos⁡x ))} =tan^(-1)⁡{√((2 cos^2⁡x/2)/(2 sin^2⁡x/2 ))} =tan^(-1)⁡|cot⁡x/2 |=tan^(-1)⁡(cot⁡x/2) ⇒y=tan^(-1)⁡{tan⁡(Ï€/2-x/2)}=Ï€/2-x/2 ∴dy/dx=0-1/2=-1/2

Q7. (d^2 x)/(dy^2 ) equals
•  ((d^2 y)/(dx^2 ))^(-1)
•  -((d^2 y)/(dx^2 ))^(-1) (dy/dx)^(-3)
•  ((d^2 y)/(dx^2 )) (dy/dx)^(-2)
•  -((d^2 y)/(dx^2 )) (dy/dx)^(-3)
Solution

Q8.If f(x)=√(1-sin⁡2x ), then f'(x) is equal to
•  -(cos⁡x+sin⁡x), for x ∈(Ï€/4,Ï€/2)
•  cos⁡x+sin⁡x, for x∈(0,Ï€/4)
•  -(cos⁡x+sin⁡x),for x∈(0,Ï€/4)
•  cos⁡x-sin⁡x,for x∈(Ï€/4,Ï€/2)
Solution
(c) f(x)=√(1-sin⁡2x )=√((cos⁡x-sin⁡x)^2 ) =|cos⁡x-sin⁡x| ={(cos⁡x-sin⁡x, for 0≤x≤Ï€/4 -(cosx-sin⁡x), for Ï€/4

Q9.

•  3/2
•  3/(4t)
•  3/2(t)
•  3t/2
Solution
(b) dy/dx=(dy|dt)/(dx|dt)=(3t^2)/2t=3/2 √x ⇒(d^2 y)/(dx^2 )=3/(4√x)=3/4t

Q10.

•  2xy/(2y-x^2)
•  xy/(y+x^2)
•  xy/(y-x^2)
• 2xy/(2+x^2/y)
Solution
(a) y=x^2+1/y ⇒y^2=x^2 y+1 ⇒2y dy/dx=y.2x+x^2 dy/dx ⇒dy/dx=2xy/(2y-x^2 )

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