alleryStart learning →d/dx (xˣ) where x > 0:
- xˣ × (1 + ln x)
- xˣ × x
- x × xˣ⁻¹
- xˣ
Show answer and solution
Correct answer: A. xˣ × (1 + ln x)
Let y = xˣ. Take ln: ln y = x ln x.
**Differentiate:** (1/y) y' = ln x + 1. So y' = xˣ × (ln x + 1) = xˣ × (1 + ln x).
KCET Limits and Continuity — practice questions — page 2
Explore 110 free Limits and Continuity questions in KCET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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Showing questions 31–60 of 110.
d/dx (xˣ) where x > 0:Read the answer and worked solution →For f(x) = sin(1/x) for x ≠ 0, what is the limit as x → 0?Read the answer and worked solution →Differentiation by chain rule for d/dx [ln(sin x)]:Read the answer and worked solution →lim_(x→0) [(1 + x)^(1/x)] =Read the answer and worked solution →Higher-order derivative: d²/dx² (x⁴) =Read the answer and worked solution →f(x) is discontinuous at x = a if which condition fails?Read the answer and worked solution →For y = x³ - 6x² + 9x, find critical points (where y' = 0):Read the answer and worked solution →By Taylor expansion around x = 0: e^x = 1 + x + ___:Read the answer and worked solution →What does the sign of f ′(x) tell you?Read the answer and worked solution →Complete the definition of a critical pointRead the answer and worked solution →Order the first derivative testRead the answer and worked solution →Match each sign change to its verdictRead the answer and worked solution →The second derivative testRead the answer and worked solution →True or false: maxima and minimaRead the answer and worked solution →Order the working rule for absolute maxima and minimaRead the answer and worked solution →Tap every point you must test for an absolute maximumRead the answer and worked solution →Local or absolute?Read the answer and worked solution →Complete the rate of change statementRead the answer and worked solution →The chain rule for related ratesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →True or false: increasing and decreasingRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Match each situation to the right toolRead the answer and worked solution →Tap every statement true of a local maximum at cRead the answer and worked solution →Reading the tests backRead the answer and worked solution →Which test is being described?Read the answer and worked solution →Assertion and reason: rates of changeRead the answer and worked solution →
Worked practice questions
For f(x) = sin(1/x) for x ≠ 0, what is the limit as x → 0?
- Does not exist (oscillates infinitely many times)
- 1
- 0
- Infinity
Show answer and solution
Correct answer: A. Does not exist (oscillates infinitely many times)
As x → 0, 1/x → ∞, and sin oscillates between -1 and 1 infinitely often.
So the function has no specific limit.
(Compare with x × sin(1/x), which DOES have limit 0 by squeeze theorem.)
Differentiation by chain rule for d/dx [ln(sin x)]:
- sin x
- tan x
- 1/sin x
- cos x / sin x = cot x
Show answer and solution
Correct answer: D. cos x / sin x = cot x
d/dx [ln(sin x)] = (1/sin x) × cos x = cot x by chain rule.
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