What is the derivative of ((3-e^{1/x})/(3+e^{1/x)}) ?

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

What is the first derivative of ((3-e^{1/x})/(3+e^{1/x)}) ?

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

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

\frac{6 e ^{\frac{1}{x}}}{x ^{2} ( 3 + e ^{\frac{1}{x}} ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

\frac{\frac{e ^{\frac{1}{x}}}{x ^{2}} ( 3 + e ^{\frac{1}{x}} ) - ( - \frac{e ^{\frac{1}{x}}}{x ^{2}} ) ( 3 - e ^{\frac{1}{x}} )}{( 3 + e ^{\frac{1}{x}} ) ^{2}} : \frac{6 e ^{\frac{1}{x}}}{x ^{2} ( 3 + e ^{\frac{1}{x}} ) ^{2}}

= \frac{6 e ^{\frac{1}{x}}}{x ^{2} ( 3 + e ^{\frac{1}{x}} ) ^{2}}

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