PRACTICE QUESTIONS ON DIFFERENTIATION FOR CA FOUNDATION

Problem 1 :

If f(x) = x^k and f^’(1) = 10, then the value of k is :

(a) 10     (b) -10     (c) 1/10      (d) None

Solution

Problem 2 :

If y = 4x³ – 7x⁴ then dy/dx is

(a)  2x(14x² – 6x)     (b) 2x(-14x² + 6x)

(c) 2x(14x² + 6x)       (d) None

Solution

Problem 3 :

If x² + y² = a², find dy/dx.

(a)  y/x        (b) – y/x       (c) –x/y       (d) x/y

Solution

Problem 4 :

Let x = at³, y = a/t³. Then dy/dx =

(a) -1/t⁶      (b) -3a/t⁶     (c) 1/3at⁶     (d) None

Solution

Problem 5 :

If y = e^3x, find y’’.

(a) 6e^3x        (b) 3e^3x        (c) 12e^3x        (d) 9e^3x

Solution

Problem 6 :

If x^m yⁿ = (x + y)^{m + n}, then find dy/dx :

(a)x/y      (b) y/x      (c) xy     (d) None

Solution

Problem 7 :

If y = log X^x then dy/dx is equal to :

(a) log ex (b) log e/x (c) log x/e (d) 1

Solution

Problem 8 :

If y = (x^1/3 - x^-1/3)³, then dy/dx is

(a) 1 + x^-2 + x^-2/3 – x^-4/3       (b) 1 + x^-2 + x^-2/3 – x^-4/3

(c) 1 – x^-2 + x^-2/3 – x^-4/3       (d) None of these

Solution

Problem 9 :

If f(x) = e^{ax^2 + bx + c} the f’(x) is

(a) e^{ax^2 + bx + c}     (b) e^{ax^2 + bx + c} (2ax + b)

(c) 2ax + b               (d) None of these

Solution

Problem 10 :

If u = 3t⁴ + 5t³ + 2t² + t + 4, then the value of du/dt at t = -1 is

(a) 0      (b) 1       (c) 2        (d) 5

Solution

Problem 11 :

If y = (x – 1) (x + 1), find d²y/dx².

(a) 2       (b) -1       (c) 0          (d) x²

Solution

Problem 12 :

The gradient of the curve

y = 2x³ – 3x² – 12x + 8 at x = 0 is

(a) -12      (b) 12      (c) 0     (d) None of these

Solution

Problem 13 :

The gradient of the curve

y = 2x³ – 5x² – 3x at x = 0 is

(a) 3      (b) -3     (c) 1/3      (d) None of these

Solution

Problem 14 :

The derivative of y = √(x + 1) is

(a) 1/√(x + 1)       (b) -1/√(x + 1)     (c) 1/2√(x + 1)

Solution

Problem 15 :

If f(x) = (x² + 1)/(x² – 1) then f’(x) is

(a) -4x/(x² – 1)²      (b) 4x/(x² – 1)²

(c) x/(x² – 1)²         (d) None of these

Solution