Question

Differentiate : √x ( x^3 + 2 )^1/3​

Answer 1

Answer:

y = 13 ( x 2 + 2 ) 3 2

Notice that  ( x 2+ 2 ) 3 2

is a function in the form

 u 3 2

. The power rule and the chain tells us that we differentiate this normally, in that the derivative of  

x 3 2

is  

3 2 x 1 2

, but when there's a function in the middle we will multiply by the derivative of the inner function. Thus, the derivative of  

u 3 2

is

 3 2 u 1 2 ⋅ dd x ( u ) .

So, we see that:

d y d x = 1 3 ( 3 2( x 2+ 2 ) 1 2 ) ⋅ d d x ( x 2 + 2 )

The power rule tells us that the derivative of

 x 2 + 2  

is  

2 x : d y d x = 1 3 ( 3 2 ) ( 2 x ) ( x 2 + 2 ) 1 2

Simplifying:

d y d x = x √ x 2 + 2

Step-by-step explanation:

Answer 2

Answer:

$\sqrt{x} { ({x}^{3} + 2) }^{ \frac{1}{3} } \\ differetiating \\ = \frac{d(\sqrt{x} { ({x}^{3} + 2) }^{ \frac{1}{3} })}{dx} \\ = \frac{d( \sqrt{x} )}{dx} \times {({x}^{3} + 2) }^{ \frac{1}{3}} + \sqrt{x} \times \frac{d({{x}^{3} + 2) }^{ \frac{1}{3}}}{dx} \\ = \frac{1}{2} \times \frac{1}{ \sqrt{x} } \times ({{x}^{3} + 2) }^{ \frac{1}{3}} + \sqrt{x} \times \frac{1}{3} \times \frac{1}{ { ({x}^{3} + 2) }^{ \frac{2}{3} } } \times \frac{d( {x}^{3} )}{dx} \\ = \frac{1}{2 \sqrt{x} } {( {x}^{3} + 2)}^{ \frac{1}{3} } + \frac{ \sqrt{x} }{3 { ({x}^{3} + 2) }^{ \frac{2}{3} } } \times 3 {x}^{2}$

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