// Numbas version: exam_results_page_options {"name": "Differentiation: Quotient Rule 6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Quotient Rule 6", "tags": [], "metadata": {"description": "
Calculating the derivative of a function of the form $\\frac{ax^n+bx}{ax^n-bx}$ using the quotient rule.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Find the derivative of \\[ \\simplify{y=({a}x^{n}+{b}x)/({a}x^{n}-{b}x)}. \\]
", "advice": "If we have a function of the form $y=\\tfrac{u(x)}{v(x)}$, to calculate its derivative we need to use the quotient rule:
\n\\[ \\dfrac{dy}{dx} = \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2}\\,.\\]
\nThis can be split up into steps:
\nFollowing this process, we must first identify $u(x)$ and $v(x)$.
\nAs \\[ \\simplify{y=({a}x^{n}+{b}x)/({a}x^{n}-{b}x)}, \\]
\nlet \\[ u(x) = \\simplify{{a}x^{n}+{b}x} \\quad \\text{and} \\quad v(x)=\\simplify{{a}x^{n}-{b}x}.\\]
\nNext, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:
\n\\[ \\dfrac{du}{dx} = \\simplify{{a*n}x^{n-1}+{b}}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=\\simplify{{a*n}x^{n-1}-{b}}.\\]
\nSubstituting these results into the quotient rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:
\n\\[ \\begin{split} \\dfrac{dy}{dx} &\\,= \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2} \\\\ \\\\&\\,=\\dfrac{(\\simplify{{a}x^{n}-{b}x}) \\times (\\simplify{{a*n}x^{n-1}+{b}}) - (\\simplify{{a}x^{n}+{b}}) \\times (\\simplify{{n*a}x^{n-1}-{b}})}{\\simplify{({a}x^{n}-{b}x)^2}}. \\end{split}\\]
\n\nSimplifying,
\n\\[ \\begin{split} \\dfrac{dy}{dx} &\\,= \\dfrac{\\simplify{({a}x^{n}-{b}x)({a*n}x^{n-1}+{b})-({a}x^{n}+{b})({n*a}x^{n-1}-{b})}}{\\simplify{({a}x^{n}-{b}x)^2}} \\\\ \\\\ &\\,=\\dfrac{\\simplify[all,!cancelTerms,!collectNumbers]{{n*a^2}x^{2*n-1}+{a*b}x^{n}-{n*a*b}x^{n}-{b^2}x-{n*a^2}x^{2*n-1}+{a*b}x^{n}-{n*a*b}x^{n}+{b^2}x}}{\\simplify{({a}x^{n}-{b}x)^2}} \\\\ \\\\ &\\,=\\dfrac{\\simplify{{2*a*b-2*n*a*b}x^{n}}}{\\simplify{({a}x^{n}-{b}x)^2}}. \\end{split} \\]
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