What is the derivative of (x^3+2x^{1/2}(x^{1/3}+5)) ?

The derivative of (x^3+2x^{1/2}(x^{1/3}+5)) is (10x^3+5x^{1/2}+15x^{1/6}+45x^{8/3})/(3x^{2/3)}

What is the first derivative of (x^3+2x^{1/2}(x^{1/3}+5)) ?

The first derivative of (x^3+2x^{1/2}(x^{1/3}+5)) is (10x^3+5x^{1/2}+15x^{1/6}+45x^{8/3})/(3x^{2/3)}

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

\frac{10 x ^{3} + 5 x ^{\frac{1}{2}} + 15 x ^{\frac{1}{6}} + 45 x ^{\frac{8}{3}}}{3 x ^{\frac{2}{3}}}

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{3} + 2 x^{\frac{1}{2}}, g = x^{\frac{1}{3}} + 5

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

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

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

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

Simplify

\frac{1 + 3 x ^{\frac{5}{2}}}{x ^{\frac{1}{2}}} (x^{\frac{1}{3}} + 5) + \frac{1}{3 x ^{\frac{2}{3}}} (x^{3} + 2 x^{\frac{1}{2}}) : \frac{10 x ^{3} + 5 x ^{\frac{1}{2}} + 15 x ^{\frac{1}{6}} + 45 x ^{\frac{8}{3}}}{3 x ^{\frac{2}{3}}}

= \frac{10 x ^{3} + 5 x ^{\frac{1}{2}} + 15 x ^{\frac{1}{6}} + 45 x ^{\frac{8}{3}}}{3 x ^{\frac{2}{3}}}

What is the derivative of (x^3+2x^{1/2}(x^{1/3}+5)) ?
The derivative of (x^3+2x^{1/2}(x^{1/3}+5)) is (10x^3+5x^{1/2}+15x^{1/6}+45x^{8/3})/(3x^{2/3)}
What is the first derivative of (x^3+2x^{1/2}(x^{1/3}+5)) ?
The first derivative of (x^3+2x^{1/2}(x^{1/3}+5)) is (10x^3+5x^{1/2}+15x^{1/6}+45x^{8/3})/(3x^{2/3)}