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

The derivative of x^{2/3}(x+4^{1/2}) is (7x+16)/(6x^{1/3)(x+4)^{1/2}}

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

The first derivative of x^{2/3}(x+4^{1/2}) is (7x+16)/(6x^{1/3)(x+4)^{1/2}}

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

\frac{7 x + 16}{6 x ^{\frac{1}{3}} ( x + 4 ) ^{\frac{1}{2}}}

Solution steps

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

Apply the Product Rule:

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

f = x^{\frac{2}{3}}, g = (x + 4)^{\frac{1}{2}}

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

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

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

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

Simplify

\frac{2}{3 x ^{\frac{1}{3}}} (x + 4)^{\frac{1}{2}} + \frac{1}{2 ( x + 4 ) ^{\frac{1}{2}}} x^{\frac{2}{3}} : \frac{7 x + 16}{6 x ^{\frac{1}{3}} ( x + 4 ) ^{\frac{1}{2}}}

= \frac{7 x + 16}{6 x ^{\frac{1}{3}} ( x + 4 ) ^{\frac{1}{2}}}

x^{2} y = (x^{3} + y^{3}) y^{'}

\int x^{2} x^{3} + 9 d x

\int e^{c}

\frac{\partial}{\partial x} (4 x - \frac{y ^{2}}{x ^{2}})

(0, 2), (2, 0)

What is the derivative of x^{2/3}(x+4^{1/2}) ?
The derivative of x^{2/3}(x+4^{1/2}) is (7x+16)/(6x^{1/3)(x+4)^{1/2}}
What is the first derivative of x^{2/3}(x+4^{1/2}) ?
The first derivative of x^{2/3}(x+4^{1/2}) is (7x+16)/(6x^{1/3)(x+4)^{1/2}}