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Longhand

Before the midterm

Differentiating a constant as if it varied

The habit: A number such as e³, π² or log 5 contains no x, so its derivative is 0.

4 practice questions are built to catch it, in 3 topics.

Practise against it: 3 questions

What it looks like

A question from exponential and trigonometric functions, with the answer this slip gives beside the right one.

Introductory2 marks · 2 min

Let f(x)=xe+ex+eef(x) = x^{e} + e^{x} + e^{e} for x>0x > 0. Find f′(x)f'(x).

With the slip
exe−1+ex+ee\displaystyle e x^{e - 1} + e^{x} + e^{e}
Worked through
f′(x)=e xe−1+ex\displaystyle f'(x) = e\,x^{e - 1} + e^{x}

ee is a number, about 15.15, with no x in it, so its derivative is 0 and it drops out of f′.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    Let f(x)=3x4−5x2+7f(x) = 3x^{4} - 5x^{2} + 7. Find f′(x)f'(x).

    Differentiation rulesIntroductory

  2. 02
    Which function satisfies both dydx=y\dfrac{dy}{dx} = y for every xx and y(0)=3y(0) = 3?

    Exponential and trigonometric functionsIntroductory

  3. 03
    Compare the equations 2y=62y = 6 and dydt=2y\dfrac{dy}{dt} = 2y, where in the second yy is an unknown function of tt. Which statement about their solutions is correct?

    Slope fields and Euler’s methodIntroductory

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