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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

f = x, g = (x^{3} + 1)^{\frac{1}{3}}

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

\frac{d x}{d x} = 1

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

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

Simplify

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

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

\frac{d}{d x} (x a^{2} + x^{2})

t^{3} \cdot sin (3 t)

\frac{\partial}{\partial t} (cos (2 t))

\int \frac{1}{x 2 + x ^{2}} d x

x = y^{2} - 1, x = 1 - y^{2}

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