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If $y=ax^n$,

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$\\frac{dy}{dx}=anx^{n-1}$ for all rational $n$.

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We'll take the following term as an example:

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$\\frac{3}{8}x^2$

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All we have to do to terms where $x$ is to a power of anything is times the coefficient of $x$ by the original power, and then take one away from the original power.

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If you are not familiar with this kind of work, these instructions may sound confusing, but it is much easier once you have seen it in practice.

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We take

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$\\frac{3}{8}x^2$

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and times $\\frac{3}{8}$ by $2$, to get

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$(\\frac{3}{8}\\times2)x^2=\\frac{6}{8}x^2=\\frac{3}{4}x^2$.

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We then subtract one from the original power, $2$.

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This gives us the final answer of

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$\\frac{3}{4}x^1=\\frac{3}{4}x$.

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Remember, don't be confused if there is no coefficient. The fact the term is there means the coefficient must be $1$, but we don't tend to write it out as, for example $1x$, we just say $x$.

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$\\simplify{({ac[0]}/{d[0]})x^3+({bc[0]}/{d[1]})x^2+({cc[0]}/{d[2]})x+({dc[0]}/{d[3]})}$

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$\\simplify{({ac[1]}/{d[4]})x^3+({bc[1]}/{d[5]})x^2+({cc[1]}/{d[6]})x+({dc[1]}/{d[7]})}$

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$\\simplify{({ac[2]}/{d[8]})x^3+({bc[2]}/{d[9]})x^2+({cc[2]}/{d[10]})x+({dc[2]}/{d[11]})}$

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Differentiate the following polynomials.

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More work on differentiation with fractional coefficients.

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