Differentiation - Past JEE - Answers

1. There is exists a function f(x) satisfying f(0) = 1, f’(0) = -1, f(x)>0 for all x , and

a. f’’(x)>0 for all x
b. -1 is less than f’’’(x)<0 for all x
c. -2≤f’’(x) ≤-1 or all x
d. f’’(x)<-2 for all x

(JEE, 1982)

Answer: (a)

Reason:

x² -x+1 = (x – ½)² + ¾ can be a solution to the f(x) satisfying f(0) = 1, f’(0) = -1, f(x)>0 for all x as f(0) = 1 and f’(0) = -1

f’’(x) = 2 and hence f’’(x)>0 for all x.

We may assume f(x) = e^-x and get a similar conclusion. But e^-x may tend to zero as x tends to infinity. Hence f(x) = x² -x+1 is more appropriate function.

2. If y = f[(2x-1)/( x²+1)] and f’(x) = sin x², then dy/dx = --------------.
(JEE 1982)

Answer: dy/dx = sin [(2x-1)/( x²+1)] ² * [2 +2x–2x²]/ ( x²+1) ²

Solution: f’(x) = sin x²
=> f’(t) = sin t²
Given problem is visualized as
y = f(t)
t = [(2x-1)/( x²+1)]
dy/dx = dy/dt*dt/dx
=> sin t² [( x²+1)(2) – (2x-1)(2x)]/ ( x²+1) ²
=> sin [(2x-1)/( x²+1)] ² * [( x²+1)(2) – (2x-1)(2x)]/ ( x²+1) ²
=> sin [(2x-1)/( x²+1)] ² * [2x²+2 –(4x²-2x)]/ ( x²+1) ²
=> sin [(2x-1)/( x²+1)] ² * [2 +2x–2x²]/ ( x²+1) ²

Differentiation - Past JEE

1. State true or false

The derivative of an even function is always an odd function (JEE 1983 Sc)

2. For the function f(x) = x/(1+e^(1/x), x≠0; and f(x) = 0 if x = o

Find the derivative from the right,

f'(0+)

and the derivative from the left
f'(0-)

(JEE 1983 sc)

3. If f(x) = log_x(ln x), then f'(x) at x = e is ______________
(JEE 1985, Sc)

4. The derivative of sec^-1{1/(2x²-1)] with respect to √(1-x²) at x = 1/2 is _______________________. (JEE 1986, Sc)

5. If y² = P(x), is a polynomial of degree 3, then

2d/dx of [y³(d²y/dx²)] equals

a. P'''(x)+P'(x)
b. P'(x)P'''(x)
c. P(x)P'''(x)
d. a constant

6. If f(x0 = |x-2| and g(x) = f[f(x)], then g'(x) ________________ for x greater than 20. (JEE 1990, Sc)

7. If y = (sin x)^{tan x}, then dy/dx is equal to

a. (sin x)^{tan x}.( 1 + sec² x. log tan x)
b. tan x.(sin x)^{tan x - 1}.cos x
c. (sin x)^{tan x}.sec² x.log sin x
d. tan x.(sin x)^{tan x-1}

(JEE 1994 Sc)

8. Let F(x) = f(x)g(x)h(x) for all real x, where f(x),g(x),h(x) are differentiable functions. At some points x ₀

F'(x₀)= 21F(x₀)

f'(x₀), = 4f(x₀),

g'(x₀), = -7g(x₀),

h'(x₀) = kh(x₀)

then k = ?

(JEE 1997, SC)

9. the left hand derivative of f(x) [x]sin (πx) at x = k, k an integer is

a. (-1)^k (k-1)π
b. (-1)^k-1 (k-1)π
c. (-1)^kkπ
d. (-1)^k-1kπ

10. The domain of the derivative of the function

f(x) = tan ^-1x if ^-1x if |x|≤1 and 1/2(|x|-1) if |x| is greater than 1 is

a. R - {-1,1)
b. R - {1}
c. R - {-1}
d. R - {0}

(JEE, 2002, Sc)