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}}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

\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}}}

d er i v a t i v e f (x) = 8 x - x^{2}

d er i v a t i v e - x^{2} + 26 x + 920

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

n = 1 \sum \infty \frac{n}{n ^{3} + 5}

\frac{d}{d x} (x 4 - x)

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}}