Derivatives — Chapter Test | Class 11 IMO

Q1

Simplify and find the derivative of f(x) = e^(logₑ x) for x > 0.

By logarithm properties, e^(logₑ x) simplifies simply to x. The derivative of x is 1.

Q2

The derivative of tan x is:

d/dx(tan x)=sec²x.

Q3

During a summer day in Jaipur, the temperature T(t) in Celsius after sunrise (t in hours) is approximated by T(t) = 25 + 2t − 0.1t². What is the rate of change of temperature 5 hours after sunrise?

dT/dt = 2 − 0.2t. At t = 5, dT/dt = 2 − 0.2(5) = 2 − 1 = 1.

Q4

The derivative of ln x is:

d/dx(ln x)=1/x.

Q5

A local manufacturer's total revenue from selling x units of ceiling fans is given by R(x) = 3x² + 36x + 5 (in ₹). Find the marginal revenue when x = 5.

Marginal Revenue is dR/dx = 6x + 36. At x = 5, MR = 6(5) + 36 = 30 + 36 = ₹ 66.

Q6

If f(x)=x³−x², then f'(2) equals:

f'(x)=3x²−2x. At x=2, value=8.

Q7

If f(x)=x² and h=0.01, then the average rate of change from x=1 to x=1+h is closest to:

[(1.01)²−1]/0.01=2.01.

Q8

Using first principles, the derivative represents:

Derivative measures instantaneous rate of change.

Q9

Find the number of points where the function f(x) = |x − 1| + |x − 2| is not differentiable.

The modulus function |x - c| has a sharp corner (non-differentiable) at x = c. This function has corners at x = 1 and x = 2.

Q10

If f(x)=2x³+x², then f'(x) is:

Derivative=6x²+2x.

Q11

The derivative of x−√x is:

Differentiate x and √x separately.

Q12

Evaluate the limit: lim(x→a) [x f(a) − a f(x)] / (x − a), given that f is differentiable at x = a.

Rewriting numerator: x f(a) - a f(a) + a f(a) - a f(x) = f(a)(x-a) - a(f(x)-f(a)). Dividing by (x-a) and taking limit yields f(a) - a f'(a).

Q13

For any positive integer n, the derivative of f(x) = xⁿ with respect to x is:

By the power rule of differentiation, d/dx (xⁿ) = n xⁿ⁻¹.

Q14

If a function f(x) has a derivative f'(x) > 0 for all x in (a, b), then in this interval the function f(x) is strictly:

A positive derivative means the slope of the tangent is positive, indicating that the function values are increasing as x increases.

Q15

A shopkeeper's profit P(x)=x²+10x. Marginal profit at x=5 is:

P'(x)=2x+10. At x=5, value=20.