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Differentiation using the product rule help


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How do you expand using square roots? I haven't come across that before

 

(x^2-3x+2)(1/2x^-1/2)+(2x-3)(sqrt x)

[(x^4/2)-(3x^2/2)+(2)][(1/2)(x^-1/2)]+[2x^-6/2][x^1/2]

 

Its a bit hard to show using text, but its just basic exponent multiplication.

 

Same base = add the exponents. Add the exponents by putting both x^ into the same denominator (in this case 2 on the bottom).

 

For example: (x^2)((1/2)(x^-1/2) -> (x^4/2)((1/2)(x^-1/2))

 

it becomes (1/2)(x^[4/2-1/2] => (1/2)(x^3/2)

 

Its much easier to understand if you write it on paper!

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Its a bit hard to show using text, but its just basic exponent multiplication.

 

Try using LaTex.

 

 

Your text will look like this instead:

 

 

 

[math](x^2-3x+2)({1}/{2}x^{-1/2})+(2x-3)(\sqrt{x})[/math]

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[math](x^2-3x+2)(\tfrac{1}{2}x^\frac{-1}{2})+(2x-3)(x^\frac{1}{2})[/math]

[math]

(x^\frac{4}{2}-3x^\frac{2}{2}+2)(\tfrac{1}{2}x^\frac{-1}{2})+(2x^\frac{-6}{2}-3)(x^\frac{1}{2})

[/math]

 

Same base = add the exponents. Add the exponents by putting both x^ into the same denominator (in this case 2 on the bottom).

 

For example:[math] (x^2)(\tfrac{1}{2}x^\frac{-1}{2}) -> (x^\frac{4}{2})(\tfrac{1}{2}x^\frac{-1}{2})[/math]

 

it becomes [math](\tfrac{1}{2}x^(\tfrac{4}{2}-\tfrac{1}{2}) => \tfrac{1}{2}x^\frac{3}{2}[/math]

 

Its much easier to understand if you write it on paper!

 

Thanks for that link for LaTeX

 

Lets try this again!

 

I hope my explaination makes a bit more sense now...

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[math]

(x^2-3x+2)(\tfrac{1}{2}x^\frac{-1}{2})+(2x-3)(x^\frac{1}{2})

[/math]

 

[math]

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

[/math]

 

Multiply the terms through

 

[math]

(\tfrac{1}{2}x^\frac{3}{2}-\tfrac{3}{2}x^\frac{1}{2}+\tfrac{2}{2}x^\frac{-1}{2})+(2x^\frac{3}{2}-3x^\frac{1}{2})

[/math]

 

Group terms with the same exponent ie:[math]x^\frac{1}{2}[/math]

 

[math]

(\tfrac{1}{2}x^\frac{3}{2}+2x^\frac{3}{2})+(-\tfrac{3}{2}x^\frac{1}{2}-3x^\frac{1}{2})+x^\frac{-1}{2}

[/math]

 

Only have one of each 'type' of x

 

[math]

(\tfrac{1}{2}x^\frac{3}{2}+\tfrac {4}{2}x^\frac{3}{2})+(-\tfrac{3}{2}x^\frac{1}{2}-\tfrac{6}{2}x^\frac{1}{2})+x^\frac{-1}{2}

[/math]

 

[math]

(\tfrac{5}{2}x^\frac{3}{2}-\tfrac{9}{2}x^\frac{1}{2}+x^\frac{-1}{2})

[/math]

 

Reduced form:

 

[math]

\tfrac{1}{2}x^\frac{-1}{2}(5x^\frac{4}{2}-9x^\frac{2}{2}+2)

[/math]

 

[math]

\tfrac{1}{2}x^\frac{-1}{2}(5x^2-9x+2)

[/math] (done here)

 

If that ended up to be -2 instead:

 

[math]

\tfrac{1}{2}x^\frac{-1}{2}(5x^2-9x-2)

[/math]

 

It can be factored like this:

 

[math]

\tfrac{1}{2}x^\frac{-1}{2}(5x+1)(x-2)

[/math]

 

I think that's right.

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