68 free MCQs on Limits and Derivatives, each with its own worked answer and explanation. Rate of change, differentiation rules, and applications
Take the timed Limits and Derivatives chapterwise test →68 practice questions on Limits and Derivatives, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Limits and Derivatives notes.
As point Q slides along the curve toward P, the secant line PQ rotates into the tangent line at P, whose slope is the derivative.
Q6.
The derivative of f(x) represents:
Q13.
At a maximum, the derivative of a function is:
Q21.
Find d/dx (x² sin x).
Q22.
Find d/dx (sin x / x).
Q24.
Find d/dx [f(g(x))] when f(u) = u² and g(x) = sin x.
Q26.
Find the critical points of f(x) = x³ - 3x² + 2.
Q27.
Second derivative (f'') > 0 at a critical point means:
Q29.
The equation of tangent to y = x² at (1,1) is:
Q31.
If f(x) = x³ - 6x² + 9x, the function is increasing when:
Q32.
The derivative of sec x is:
Q33.
d/dx (x<sup>x</sup>) =
Q34.
The derivative of sin⁻¹(x) is:
Q38.
Rolle's theorem requires f to be:
Q41.
Evaluate: lim(x→0) [sin(x)/x]<sup>1/x²</sup> using limits.
Q43.
If y = (sin x)<sup>tan x</sup>, find dy/dx.
Q44.
The Mean Value Theorem states that for f on [a,b]:
Q46.
The inflection point of f(x) = x⁴ - 4x³ is at:
Q47.
Differentiate: y = cos⁻¹(2x²-1) with respect to x.
Q50.
Taylor series expansion of eˣ around x=0 gives the n-th derivative value at 0 as:
Q51.
Find the equation of normal to curve xy = 4 at point (2,2).
Q52.
Newton-Leibniz rule for d/dx [integral from a to g(x) of f(t) dt] =
Q53.
If f''(a) > 0 and f'(a) = 0 and f''(a) exists, then f(a) is:
Q54.
Differentiate: y = sin(cos(tan x))
Q55.
If x = a cos t, y = a sin t (parametric), dy/dx =
Q56.
Cauchy Mean Value Theorem applies to functions f and g. It states: there exists c in (a,b) such that:
Q60.
The derivative of sin x with respect to x is:
Q61.
The limit as n → ∞ of (1/n)·[(n+1)(n+2)...(2n)]<sup>1/n</sup> is:
Q62.
The limit as x → 0 of (cos x)<sup>1/x²</sup> is:
Q64.
The limit as x → 0 of (e<sup>2x</sup> − 1) / ln(1 + 3x) is:
Q66.
The limit as x → 0 of (1 + 2x)<sup>1/x</sup> is:
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