What is the derivative of x(4x+1^{1/3}) ?

The derivative of x(4x+1^{1/3}) is (16x+3)/(3(4x+1)^{2/3)}

What is the first derivative of x(4x+1^{1/3}) ?

The first derivative of x(4x+1^{1/3}) is (16x+3)/(3(4x+1)^{2/3)}

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

\frac{16 x + 3}{3 ( 4 x + 1 ) ^{\frac{2}{3}}}

Solution steps

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

Apply the Product Rule:

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

f = x, g = (4 x + 1)^{\frac{1}{3}}

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

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

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

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

Simplify

1 \cdot (4 x + 1)^{\frac{1}{3}} + \frac{4}{3 ( 4 x + 1 ) ^{\frac{2}{3}}} x : \frac{16 x + 3}{3 ( 4 x + 1 ) ^{\frac{2}{3}}}

= \frac{16 x + 3}{3 ( 4 x + 1 ) ^{\frac{2}{3}}}

\int \frac{t}{t - 2} d t

\int \frac{1}{x - c} d x

\int \frac{x ^{4} - 2 x ^{2} + 1}{x ^{3}} d x

x \to 1 - lim (4 x - 5)

\frac{2}{3 x}

What is the derivative of x(4x+1^{1/3}) ?
The derivative of x(4x+1^{1/3}) is (16x+3)/(3(4x+1)^{2/3)}
What is the first derivative of x(4x+1^{1/3}) ?
The first derivative of x(4x+1^{1/3}) is (16x+3)/(3(4x+1)^{2/3)}