# How do you differentiate #(4x-3)/sqrt (2x^2+1)#?

Finally we get

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To differentiate the function ( \frac{{4x - 3}}{{\sqrt{{2x^2 + 1}}}} ), you can use the quotient rule, which states that if you have a function of the form ( \frac{{f(x)}}{{g(x)}} ), its derivative is given by:

[ \frac{{f'(x)g(x) - f(x)g'(x)}}{{[g(x)]^2}} ]

where ( f'(x) ) represents the derivative of ( f(x) ) and ( g'(x) ) represents the derivative of ( g(x) ).

In this case:

[ f(x) = 4x - 3 ] [ g(x) = \sqrt{{2x^2 + 1}} ]

To find ( f'(x) ) and ( g'(x) ):

[ f'(x) = 4 ] [ g'(x) = \frac{{4x}}{{2\sqrt{{2x^2 + 1}}}} ]

Now, using the quotient rule:

[ \frac{{d}}{{dx}}\left(\frac{{4x - 3}}{{\sqrt{{2x^2 + 1}}}}\right) = \frac{{(4)(\sqrt{{2x^2 + 1}}) - (4x - 3)\left(\frac{{4x}}{{2\sqrt{{2x^2 + 1}}}}\right)}}{{(2x^2 + 1)}} ]

Simplify the expression to get the final derivative.

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