// Numbas version: exam_results_page_options {"name": "Differentiation 1 - Basic Polynomial Expressions ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["ac", "bc", "cc", "dc", "ec", "fc", "ap", "bp", "cp", "dp"], "name": "Differentiation 1 - Basic Polynomial Expressions ", "tags": [], "advice": "

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 one of the terms from Part a as an example:

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$\\var{cc[0]}x^\\var{cp}$

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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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$\\var{cc[0]}x^\\var{cp}$

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and times $\\var{cc[0]}$ by $\\var{cp}$, to get

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$(\\var{cc[0]}*\\var{cp})x^\\var{cp}=\\simplify{{cc[0]}*{cp}x^{cp}}$.

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We then subtract one from the original power, $\\var{cp}$.

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This gives us the final answer of

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$\\simplify{{cc[0]}*{cp}x^{cp-1}}$.

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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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$\\simplify[basic,zeroterm,zerofactor,unitfactor]{{ac[1]}x^{ap}+{bc[1]}x^{bp}+{cc[1]}x^{cp}+{dc[1]}x^{dp}+{ec[1]}x+{fc[1]}}$

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$\\frac{dy}{dx}=$ [[0]]

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Differentiate the following polynomials.

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Note: some questions may not include all the possible terms.

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A basic introduction to differentiation

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