What is the derivative of \sqrt[3]{x+sqrt(x^5)} ?

The derivative of \sqrt[3]{x+sqrt(x^5)} is (2+5x^{3/2})/(6(x+x^2sqrt(x))^{2/3)}

What is the first derivative of \sqrt[3]{x+sqrt(x^5)} ?

The first derivative of \sqrt[3]{x+sqrt(x^5)} is (2+5x^{3/2})/(6(x+x^2sqrt(x))^{2/3)}

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

Solution steps

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{3 ( x + x ^{5} ) ^{\frac{2}{3}}} (1 + \frac{5 x ^{\frac{3}{2}}}{2}) : \frac{2 + 5 x ^{\frac{3}{2}}}{6 ( x + x ^{2} x ) ^{\frac{2}{3}}}

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

\frac{2 d y}{d x} = y^{2}

\frac{d}{d x} (3 - ln (6 x) - \frac{x ^{2}}{12} + x)

\int \frac{1 + ln ( x )}{x} d x

3 t^{2} + 25

x \to - 4 - lim (\frac{1}{x ^{2} - 16})

What is the derivative of \sqrt[3]{x+sqrt(x^5)} ?
The derivative of \sqrt[3]{x+sqrt(x^5)} is (2+5x^{3/2})/(6(x+x^2sqrt(x))^{2/3)}
What is the first derivative of \sqrt[3]{x+sqrt(x^5)} ?
The first derivative of \sqrt[3]{x+sqrt(x^5)} is (2+5x^{3/2})/(6(x+x^2sqrt(x))^{2/3)}