What is the derivative of (e^x+1/(e^{2x)-5e^x+6}) ?

The derivative of (e^x+1/(e^{2x)-5e^x+6}) is (11e^x-e^{3x}-2e^{2x})/((e^{2x)-5e^x+6)^2}

What is the first derivative of (e^x+1/(e^{2x)-5e^x+6}) ?

The first derivative of (e^x+1/(e^{2x)-5e^x+6}) is (11e^x-e^{3x}-2e^{2x})/((e^{2x)-5e^x+6)^2}

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\frac{d}{d x} (\frac{e ^{x} + 1}{e ^{2 x} - 5 e ^{x} + 6})

\frac{11 e ^{x} - e ^{3 x} - 2 e ^{2 x}}{( e ^{2 x} - 5 e ^{x} + 6 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{e ^{x} + 1}{e ^{2 x} - 5 e ^{x} + 6})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( e ^{x} + 1 ) ( e ^{2 x} - 5 e ^{x} + 6 ) - \frac{d}{d x} ( e ^{2 x} - 5 e ^{x} + 6 ) ( e ^{x} + 1 )}{( e ^{2 x} - 5 e ^{x} + 6 ) ^{2}}

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

\frac{d}{d x} (e^{2 x} - 5 e^{x} + 6) = - 5 e^{x} + 2 e^{2 x}

= \frac{e ^{x} ( e ^{2 x} - 5 e ^{x} + 6 ) - ( - 5 e ^{x} + 2 e ^{2 x} ) ( e ^{x} + 1 )}{( e ^{2 x} - 5 e ^{x} + 6 ) ^{2}}

Expand

e^{x} (e^{2 x} - 5 e^{x} + 6) - (- 5 e^{x} + 2 e^{2 x}) (e^{x} + 1) : 11 e^{x} - e^{3 x} - 2 e^{2 x}

= \frac{11 e ^{x} - e ^{3 x} - 2 e ^{2 x}}{( e ^{2 x} - 5 e ^{x} + 6 ) ^{2}}

\frac{\partial}{\partial x} (2 x + 3 x^{2} y)

(x e^{- 3 x})^{^{'}}

\int_{0}^{2} x^{3} - 3 x^{2} + 4 d x

y^{^{'}} - \frac{5 y}{x} = 3 x^{4}, y (1) = 5

2 x \frac{d y}{d x} = (cos^{2} (y))

What is the derivative of (e^x+1/(e^{2x)-5e^x+6}) ?
The derivative of (e^x+1/(e^{2x)-5e^x+6}) is (11e^x-e^{3x}-2e^{2x})/((e^{2x)-5e^x+6)^2}
What is the first derivative of (e^x+1/(e^{2x)-5e^x+6}) ?
The first derivative of (e^x+1/(e^{2x)-5e^x+6}) is (11e^x-e^{3x}-2e^{2x})/((e^{2x)-5e^x+6)^2}