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  1. 024Differentiate f(x) = x to the power x, and find where it reaches its minimum for positive x.Mental maths and number senseCoreScotiabankToronto · 2026

    Try it first

    What is the derivative of x to the x?

    Show the worked solution

    f'(x) = x to the x times (ln x + 1), and the minimum is at x = 1/e, about 0.368, where f is about 0.692. Take logs: ln f = x ln x. Differentiating, f'/f = ln x + 1, so f' = x to the x (ln x + 1). The derivative is zero when ln x = -1, that is x = 1/e, negative before it and positive after, so this is a minimum.

    Why do both standard rules fail?

    The power rule, n x to the (n - 1), treats the exponent as fixed; the exponential rule, a to the x times ln a, treats the base as fixed. In x to the x both the base and the exponent move, so neither rule applies on its own, and each one gives half of the right answer. Indeed the correct derivative is the sum of the two: x times x to the (x - 1), which is x to the x, plus x to the x ln x. That sum is a quick check on your final answer.

    Take logs first: the minimum sits at x = 1/e either waymin 0.692 at x = 1/e = 0.3682 to the 2 = 40121234f(x) = x to the xmin -1/e = -0.36812ln f(x) = x ln xd/dx of x ln x = ln x + 1f'(x) = x to the x (ln x + 1)
    The curve x to the x falls from near 1 at zero to a minimum of 0.692 at x = 1/e and then rises to 4 at x = 2, and its logarithm x ln x has its minimum at the same point, which is why taking logs first is safe.

    How does taking logs make it routine?

    Think of converting a messy multiplication into addition before doing it, the way a slide rule does. Write ln f = x ln x; the right-hand side is a product of two simple functions, and the product rule gives ln x + x times 1/x = ln x + 1. The left side differentiates to f'/f by the chain rule, so f' = f (ln x + 1). This is logarithmic differentiationDifferentiating the logarithm of a function instead of the function, then multiplying back; useful when the variable sits in an exponent., and it works for any function of the form g(x) to the h(x).

    The relationship
    f(x)=exln⁡x  ⇒  f′(x)=exln⁡x(ln⁡x+1)=xx(ln⁡x+1)=0  ⟺  x=e−1≈0.368f(x) = e^{x\ln x} \;\Rightarrow\; f'(x) = e^{x\ln x}(\ln x + 1) = x^x(\ln x + 1) = 0 \iff x = e^{-1} \approx 0.368
    e^{x ln x}x to the x rewritten with a fixed base
    ln x + 1the derivative of x ln x
    e^{-1}where ln x = -1, the minimum
    What it says in wordsRewrite with base e, differentiate the exponent, and set it to zero.

    How do you confirm it is a minimum and state the value?

    Check the sign of ln x + 1, since x to the x is always positive. For x below 1/e, ln x is below -1 and the slope is negative; above 1/e it is positive, so the function falls and then rises: a minimum. The value is (1/e) to the (1/e) = e to the (-1/e), about 0.692. Also say what happens at the edges: as x shrinks towards zero, x ln x tends to zero, so x to the x tends to 1, and at x = 1 it is exactly 1 again.

    Where candidates lose it

    The fast wrong answer applies the power rule, x times x to the (x - 1), which is just x to the x. It treats the exponent as a constant, and candidates who give it usually do so in the first three seconds.

    The second loss is finding x = 1/e and stopping. The question asks for the minimum, so check the sign change and give the value, e to the (-1/e), about 0.692, together with the behaviour near zero.

    What the interviewer asks next

    • Differentiate x to the (x to the x).
    • What is the limit of x to the x as x approaches 0 from above, and why?
    • Which is larger, e to the pi or pi to the e, and how does x to the (1/x) settle it?

    Asked at Scotiabank, Quant, Toronto, 2026 (Wall Street Oasis): technical questions covering calculus (including derivatives of standard functions)

  2. 061A Rs 1,000 crore fund charges a 2% management fee and 20% of gains, with the performance fee taken on the gain left after the management fee. In a year with a 10% gross return, the manager cuts the management fee to 1%. What performance fee keeps the manager's revenue unchanged?Mental maths and number senseCoreTwo SigmaNew York · 2026

    Try it first

    Which performance fee keeps revenue at the old level?

    Show the worked solution

    About 28.9%, roughly 29%. Under 2 and 20 the manager earns Rs 20 crore of management fee plus 20% of the remaining Rs 80 crore gain, Rs 36 crore in all. At 1% the management fee is Rs 10 crore and the gain left is Rs 90 crore, so the performance fee must bring in Rs 26 crore: 26 / 90 = 28.9%. It only balances at a 10% return.

    Why is the answer not simply 30%?

    Think of a shopkeeper who takes a fixed rent from a stall plus a fifth of whatever the stall-holder clears after paying that rent. If the rent is halved, the stall-holder clears more, and the shopkeeper's fifth is a fifth of a larger amount. Cutting the management fee enlarges the gain on which the performance fee is charged, so part of the lost management revenue comes back automatically. Here the old fee of 20% on Rs 90 crore earns Rs 18 crore, up from Rs 16 crore, and the true shortfall is Rs 8 crore, not Rs 10 crore. Adding 10 points to get 30% repays the shortfall twice over in part.

    The fee swap only balances at one return: here, 10% grossRevenue at a 10% gross return, Rs crore2016362% + 20%before1018-8281% + 20%fee cut only1026361% + 28.9%re-pricedmanagement feeperformance feeRevenue at other gross returns2040600%5%10%15%20%cross at 10%: 36 eachnew terms: 64.92 and 20: 56below 10%:manager earns lessGross return on the fund
    At a 10% gross return the old terms earn Rs 36 crore, the fee cut alone earns Rs 28 crore, and a performance fee of 28.9% restores Rs 36 crore; at any other return the two sets of terms pay differently, crossing only at 10%.

    How do you set it up in one line?

    Write revenue as management fee plus performance rate times the gain after the management fee, and hold it equal. The new rate is the old revenue less the new management fee, divided by the new fee base: (36 - 10) / 90 = 28.9%. The investor's position is the mirror image: gross gain Rs 100 crore less Rs 36 crore of fees leaves Rs 64 crore, a 6.4% net return under either set of terms, which is a good check that you have kept the total fixed.

    The relationship
    m0A+p0(G−m0A)=m1A+p1(G−m1A)  ⇒  p1=20+0.2×80−1090=2690≈28.9%m_0 A + p_0 (G - m_0 A) = m_1 A + p_1 (G - m_1 A) \;\Rightarrow\; p_1 = \frac{20 + 0.2 \times 80 - 10}{90} = \frac{26}{90} \approx 28.9\%
    Aassets under management, Rs 1,000 crore
    Gthe gross gain, 10% of A, Rs 100 crore
    m0, m1the old and new management fee rates, 2% and 1%
    p0, p1the old and new performance fee rates
    What it says in wordsHold total revenue fixed and solve for the performance rate on the new, larger fee base.

    What does the swap change, even when revenue matches?

    The match holds at one return only. Swapping management fee for performance fee moves risk onto the manager: less is guaranteed, more depends on the year. At a 5% gross return the old terms pay Rs 26 crore and the new ones Rs 21.6 crore; at 20% the old terms pay Rs 56 crore and the new ones Rs 64.9 crore. State the convention too: if the performance fee were charged on the gross gain instead, the old revenue is Rs 40 crore and the new rate comes to exactly 30%. Say which convention you assume before giving a number.

    Where candidates lose it

    The fast wrong answers are 40%, from doubling the performance fee because the management fee halved, and 30%, from adding back ten points. Both forget that the performance fee is charged on a base the management fee itself shrinks.

    The second loss is giving one number without the conditions. A fee trade-off is always at an assumed return and an assumed fee base; say both and offer the sensitivity, because the interviewer's next question is what happens in a bad year.

    What the interviewer asks next

    • At what gross return would the manager prefer the new terms to the old?
    • How does a hurdle rate of 5% change the new performance fee?
    • Why might investors prefer 1 and 29 to 2 and 20 even though revenue matches at 10%?

    Asked at Two Sigma, Equity Capital Markets, New York, 2026 (Wall Street Oasis): the question was regarding how hedge funds operate, the 2/20 rule, and if one part of this equation changed