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Calculating the derivative of a function of the form $\\frac{ax^n+bx^m}{cx^p+d}$ using the quotient rule.

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Find the derivative of \\[ \\simplify{y=({a}x^{n}+{b}x^{m})/({c}x^{p}+{d})}. \\]

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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:

\\[ \\dfrac{dy}{dx} = \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2}\\,.\\]

This can be split up into steps:

Identify the functions $u(x)$ and $v(x)$;
Calculate their derivatives $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$;
Substitute these into the formula for the quotient rule to obtain an expression for $\\tfrac{dy}{dx}$;
Simplify $\\tfrac{dy}{dx}$ where possible.

Following this process, we must first identify $u(x)$ and $v(x)$.

As \\[ \\simplify{y=({a}x^{n}+{b}x^{m})/({c}x^{p}+{d})}, \\]

let \\[ u(x) = \\simplify{{a}x^{n}+{b}x^{m}} \\quad \\text{and} \\quad v(x)=\\simplify{{c}x^{p}+{d}}.\\]

Next, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:

\\[ \\dfrac{du}{dx} = \\simplify{{a*n}x^{n-1}+{b*m}x^{m-1}}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=\\simplify{{c*p}x^{p-1}}.\\]

Substituting these results into the quotient rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:

\\[ \\begin{split} \\dfrac{dy}{dx} &\\,= \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2} \\\\ \\\\&\\,=\\dfrac{(\\simplify{{c}x^{p}+{d}}) \\times (\\simplify{{a*n}x^{n-1}+{b*m}x^{m-1}}) - (\\simplify{{a}x^{n}+{b}x^{m}}) \\times \\simplify{{c*p}x^{p-1}}}{\\simplify{({c}x^{p}+{d})^2}}. \\end{split}\\]

Simplifying,

\\[ \\begin{split} \\dfrac{dy}{dx} &\\,=\\dfrac{(\\simplify{{c}x^{p}+{d}})(\\simplify{{a*n}x^{n-1}+{b*m}x^{m-1}}) -\\simplify{{c*p}x^{p-1}}(\\simplify{{a}x^{n}+{b}x^{m}})}{\\simplify{({c}x^{p}+{d})^2}} \\\\ \\\\&\\,= \\dfrac{\\simplify[all,!cancelTerms]{{c*n*a}x^{p+n-1}+{c*m*b}x^{p+m-1}+{d*n*a}x^{n-1}+{d*b*m}x^{m-1}-{a*p*c}x^{n+p-1}-{b*p*c}x^{p+m-1}}}{\\simplify{({c}x^{p}+{d})^2}}\\\\ \\\\ &\\,= \\dfrac{\\simplify{{c*n*a-a*p*c}x^{p+n-1}+{c*m*b-b*p*c}x^{p+m-1}+{d*n*a}x^{n-1}+{d*b*m}x^{m-1}}}{\\simplify{({c}x^{p}+{d})^2}} \\end{split} \\]

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$\\dfrac{dy}{dx}=$[[0]]

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