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Find the derivative of the function using chain rule and general power rule `y=x...

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rmunoz90 | Student, Undergraduate | Salutatorian

Posted March 12, 2013 at 4:06 PM via web

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Find the derivative of the function using chain rule and general power rule

`y=x sqrt(2x+3)`

chain rule=

general power rule=

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted March 12, 2013 at 5:19 PM (Answer #1)

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Chain rule

`f(g(x))'=f'(g(x))cdot g'(x)`

E.g.

`(sin(x^2))'=cos(x^2)cdot2x`

Here `f(x)=sin(x)`  and `g(x)=x^2`.

General power rule

`(x^n)'=nx^(n-1)`

E.g.

`(x^3)'=3x^2` 

We will also need product rule

`(f cdot g)'=f' cdot g+f cdot g'`

Let's now differentiate our function `y`.

`y'=(x sqrt(2x+3))'=`

We first use product rule.

`x'sqrt(2x+3)+x(sqrt(2x+3))'=`                               (1)

Now we use general power rule for

`x'=1` 

and then we use chain rule for

`(sqrt(2x+3))'=1/2sqrt(2x+3)cdot2=1/sqrt(2x+3)`.`` 

Now we put that into (1) to get `y'.`  

Thus the solution is:

`y'=sqrt(2x+3)+x/sqrt(2x+3)`

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