What is the derivative of (x^2-2e^x)/(x+3e^x) ?

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

What is the first derivative of (x^2-2e^x)/(x+3e^x) ?

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

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derivative of

\frac{x ^{2} - 2 e ^{x}}{x + 3 e ^{x}}

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

Solution steps

\frac{d}{d x} (\frac{x ^{2} - 2 e ^{x}}{x + 3 e ^{x}})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x ^{2} - 2 e ^{x} ) ( x + 3 e ^{x} ) - \frac{d}{d x} ( x + 3 e ^{x} ) ( x ^{2} - 2 e ^{x} )}{( x + 3 e ^{x} ) ^{2}}

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

\frac{d}{d x} (x + 3 e^{x}) = 1 + 3 e^{x}

= \frac{( 2 x - 2 e ^{x} ) ( x + 3 e ^{x} ) - ( 1 + 3 e ^{x} ) ( x ^{2} - 2 e ^{x} )}{( x + 3 e ^{x} ) ^{2}}

Expand

(2 x - 2 e^{x}) (x + 3 e^{x}) - (1 + 3 e^{x}) (x^{2} - 2 e^{x}) : x^{2} - 3 e^{x} x^{2} + 4 e^{x} x + 2 e^{x}

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

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