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Derivatives

Theory and Formula




Question



Q1. The side of an equilateral triangle is increasing at the rate of 2cm/s . At what rate is its area increasing , when the side of the triangle is 25 cm ?

(a) 25√3 cm2/s (b) 30√3 cm2/s (c) 15√3 cm2/s (d) 50√3 cm2/s

Solution.
25√3 cm2/s
Solution

Q2. If x3 - 2x2y2 + 6x + y = 10, then dy/dx at x = 1 and y = 1 is.

(a) 1/3 (b) 5/4 (c) 5/3 (d) 2/3

Solution.
(c) 5/3
Solution

Q3. If x = logθ , y = eθ, find dy/dx

(a) eθ (b) θ.eθ (c) θ (d) 1

Solution.
(b) θ.eθ
Solution

Q4. The volume of a sphere is increasing at the rate of 8 cm3/s . find the rate at which its surface area is increasing when the radius of the sphere is 16 cm.

(a) 1cm2/sec (b) 2cm2/sec (c) 3cm2/sec (d) 4cm2/sec

Solution.
1cm2/sec
Solution

Q5. If log(x/y) = x + y , find dy/dx.

(a)[y(1-x)]/[x(1+y)] (b) 2y (c) 2x (d) y/x

Solution.
(a)[y(1-x)]/[x(1+y)]
Solution

Q6. If y = xx, find dy/dx at x = 1
(a) 1 (b) 2 (c) 3 (d) 4

Answer.
(a) 1
Solution

Q7. If f(x)= y = sin-1 (
6x / 1 + 9x2
), find f '(1) .
(a) 3/5 (b) 1/5 (c) 3 (d) 5

Answer.
(a) 3/5
Solution

Q8. Find
dy/dx
, If y = log3(log x)
(a) log x (b)
1 / log x . log 3
(c)
1 / xlog3 logx
(d) log3

Answer.
(c)
1 / xlog3 logx
Solution

Q9. If y = functionfunction, then for finding dy/dx we follow

(a) Product rule (b) Quotient rule (c) Logarithmic rule (d) None of these

Solution.
(c) Logarithmic rule

Note: In this take log both side with base e, then simplify using rule of logarithm and then differentiate with respect to "x" on both sides.
Solution

Q10. Find d2y/dx2 , if y = x2

Solution.
y = 2
Solution