What is the derivative of e^x((2x+5)^2)/(x-2) ?

The derivative of e^x((2x+5)^2)/(x-2) is (4e^xx^3+16e^xx^2-31e^xx-115e^x)/((x-2)^2)

What is the first derivative of e^x((2x+5)^2)/(x-2) ?

The first derivative of e^x((2x+5)^2)/(x-2) is (4e^xx^3+16e^xx^2-31e^xx-115e^x)/((x-2)^2)

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derivative of

e^{x} \frac{( 2 x + 5 ) ^{2}}{x - 2}

\frac{4 e ^{x} x ^{3} + 16 e ^{x} x ^{2} - 31 e ^{x} x - 115 e ^{x}}{( x - 2 ) ^{2}}

Solution steps

\frac{d}{d x} (e^{x} \frac{( 2 x + 5 ) ^{2}}{x - 2})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{x}, g = \frac{( 2 x + 5 ) ^{2}}{x - 2}

= \frac{d}{d x} (e^{x}) \frac{( 2 x + 5 ) ^{2}}{x - 2} + \frac{d}{d x} (\frac{( 2 x + 5 ) ^{2}}{x - 2}) e^{x}

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} (\frac{( 2 x + 5 ) ^{2}}{x - 2}) = \frac{4 x ^{2} - 16 x - 65}{( x - 2 ) ^{2}}

= e^{x} \frac{( 2 x + 5 ) ^{2}}{x - 2} + \frac{4 x ^{2} - 16 x - 65}{( x - 2 ) ^{2}} e^{x}

Simplify

e^{x} \frac{( 2 x + 5 ) ^{2}}{x - 2} + \frac{4 x ^{2} - 16 x - 65}{( x - 2 ) ^{2}} e^{x} : \frac{4 e ^{x} x ^{3} + 16 e ^{x} x ^{2} - 31 e ^{x} x - 115 e ^{x}}{( x - 2 ) ^{2}}

= \frac{4 e ^{x} x ^{3} + 16 e ^{x} x ^{2} - 31 e ^{x} x - 115 e ^{x}}{( x - 2 ) ^{2}}

What is the derivative of e^x((2x+5)^2)/(x-2) ?
The derivative of e^x((2x+5)^2)/(x-2) is (4e^xx^3+16e^xx^2-31e^xx-115e^x)/((x-2)^2)
What is the first derivative of e^x((2x+5)^2)/(x-2) ?
The first derivative of e^x((2x+5)^2)/(x-2) is (4e^xx^3+16e^xx^2-31e^xx-115e^x)/((x-2)^2)