// Numbas version: finer_feedback_settings {"name": "Introduction in worksheet theme", "metadata": {"description": "This exam shows that the worksheet theme now includes the introduction text in each copy of the exam.
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The derivative of $x^n$ is given by the following:

\\[ \\frac{\\mathrm{d}}{\\mathrm{d}x}(x^n) = n \\times x^{n-1} \\]

Enter the derivatives of each of the three terms in $f(x)$:

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$\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[2]}*x^{powers[2]}}) =$

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$\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[1]}*x^{powers[1]}}) =$

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$\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[0]}*x^{powers[0]}}) =$

Differentiate the following function.

\\[ f(x) = \\simplify[all,!noLeadingMinus]{ {coefficients[2]}*x^{powers[2]} + {coefficients[1]}*x^{powers[1]} + {coefficients[0]}*x^{powers[0]} } \\]

$\\frac{\\mathrm{d}f}{\\mathrm{d}x} = $ [[0]]

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Numbas is really good at creating and marking randomised maths questions. In this question, you're given a random polynomial to differentiate.

Notice how Numbas automatically simplifies the mathematical expressions so they look as if a human wrote them.

See this question in the public editor

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The derivative of $x^n$ is given by the following:

\\[ \\frac{\\mathrm{d}}{\\mathrm{d}x}(x^n) = n \\times x^{n-1} \\]

We can compute the derivative of $f(x)$ by computing the derivatives of each of the three terms, and then adding them together.

\\begin{align}
\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[2]}*x^{powers[2]}}) &= \\simplify[basic]{{powers[2]}*{coefficients[2]}*x^({powers[2]}-1)} \\\\
&= \\simplify{{coefficients[2]*powers[2]}*x^{powers[2]-1}}
\\end{align}

\\begin{align}
\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[1]}*x^{powers[1]}}) &= \\simplify[basic]{{powers[1]}*{coefficients[1]}*x^({powers[1]}-1)} \\\\
&= \\simplify{{coefficients[1]*powers[1]}*x^{powers[1]-1}}
\\end{align}

The derivative of a constant is $0$. So,

\\[ \\frac{\\mathrm{d}}{\\mathrm{d}x}(\\var{coefficients[0]}) = 0 \\]

\\begin{align}
\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[0]}*x^{powers[0]}}) &= \\simplify[basic]{{powers[0]}*{coefficients[0]}*x^({powers[0]}-1)} \\\\
&= \\simplify{{coefficients[0]*powers[0]}*x^{powers[0]-1}}
\\end{align}

Hence,

\\[ \\frac{\\mathrm{d}f}{\\mathrm{d}x} = \\simplify{ {coefficients[2]*powers[2]}*x^{powers[2]-1} + {coefficients[1]*powers[1]}*x^{powers[1]-1} + {coefficients[0]*powers[0]}*x^{powers[0]-1} } \\]

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Write your name:

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