What is the derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) ?

The derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) is (e^{(5x)/4}(5(x^2+1)-2x))/(4(x^2+1)^{5/4)}

What is the first derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) ?

The first derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) is (e^{(5x)/4}(5(x^2+1)-2x))/(4(x^2+1)^{5/4)}

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\frac{e ^{\frac{5 x}{4}} ( 5 ( x ^{2} + 1 ) - 2 x )}{4 ( x ^{2} + 1 ) ^{\frac{5}{4}}}

Solution steps

Apply the chain rule:

\frac{1}{4 ( \frac{e ^{5 x}}{x ^{2} + 1} ) ^{\frac{3}{4}}} \frac{d}{d x} (\frac{e ^{5 x}}{x ^{2} + 1})

= \frac{1}{4 ( \frac{e ^{5 x}}{x ^{2} + 1} ) ^{\frac{3}{4}}} \frac{d}{d x} (\frac{e ^{5 x}}{x ^{2} + 1})

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

= \frac{1}{4 ( \frac{e ^{5 x}}{x ^{2} + 1} ) ^{\frac{3}{4}}} \cdot \frac{e ^{5 x} \cdot 5 ( x ^{2} + 1 ) - 2 x e ^{5 x}}{( x ^{2} + 1 ) ^{2}}

Simplify

\frac{1}{4 ( \frac{e ^{5 x}}{x ^{2} + 1} ) ^{\frac{3}{4}}} \cdot \frac{e ^{5 x} \cdot 5 ( x ^{2} + 1 ) - 2 x e ^{5 x}}{( x ^{2} + 1 ) ^{2}} : \frac{e ^{\frac{5 x}{4}} ( 5 ( x ^{2} + 1 ) - 2 x )}{4 ( x ^{2} + 1 ) ^{\frac{5}{4}}}

= \frac{e ^{\frac{5 x}{4}} ( 5 ( x ^{2} + 1 ) - 2 x )}{4 ( x ^{2} + 1 ) ^{\frac{5}{4}}}

What is the derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) ?
The derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) is (e^{(5x)/4}(5(x^2+1)-2x))/(4(x^2+1)^{5/4)}
What is the first derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) ?
The first derivative of \sqrt[4]{(e^{5x}/(x^2+1)}) is (e^{(5x)/4}(5(x^2+1)-2x))/(4(x^2+1)^{5/4)}