What is the derivative of the function f(x)=\frac{3x-5}{5x+2}?

Layla-Rose Ellison 2022-02-28 Answered
What is the derivative of the function
$f\left(x\right)=\frac{3x-5}{5x+2}$?
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Step 1
The quotient rule states that
if $y=\frac{f\left(x\right)}{g\left(x\right)}$, then $\frac{dy}{dx}=\frac{{f}^{\prime }\left(x\right)g\left(x\right)-{g}^{\prime }\left(x\right)f\left(x\right)}{{\left(g\left(x\right)\right)}^{2}}$
Now if we apply this to the given function
${f}^{\prime }\left(x\right)=\frac{3\left(5x+2\right)-5\left(3x-5\right)}{{\left(5x+2\right)}^{2}}$
${f}^{\prime }\left(x\right)=\frac{15x+6-15x+25}{{\left(5x+2\right)}^{2}}$
${f}^{\prime }\left(x\right)=\frac{31}{{\left(5x+2\right)}^{2}}$
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twaza7fl
Step 1
Quotient Rule: ${\left(\frac{u}{v}\right)}^{\prime }=\frac{v{u}^{\prime }-u{v}^{\prime }}{{v}^{2}}$
${\left(\frac{u}{v}\right)}^{\prime }=\frac{\left(5x+2\right)\left(3\right)-\left(3x-5\right)\left(5\right)}{{\left(5x+2\right)}^{2}}$
$=\frac{15x+6-\left(15x-25\right)}{{\left(5x+2\right)}^{2}}$
$=\frac{31}{{\left(5x+2\right)}^{2}}$
If you are enrolled in a calculus course, you really need to review these elementary differentiation rules.