# How do you differentiate #y=(4x^5+x^2+4)/(5x^2-2)# using the quotient rule?

This is what you should end up when you derive the equation.

Then distribute the top and combine like terms.

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To differentiate ( y = \frac{4x^5 + x^2 + 4}{5x^2 - 2} ) using the quotient rule, the formula is:

[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{(g(x))^2} ]

Where ( f(x) = 4x^5 + x^2 + 4 ) and ( g(x) = 5x^2 - 2 ).

Now, differentiate ( f(x) ) and ( g(x) ) to get ( f'(x) ) and ( g'(x) ), respectively. Then, substitute into the quotient rule formula and simplify.

[ f'(x) = 20x^4 + 2x ]

[ g'(x) = 10x ]

Now, substitute these into the quotient rule formula:

[ \frac{(5x^2 - 2)(20x^4 + 2x) - (4x^5 + x^2 + 4)(10x)}{(5x^2 - 2)^2} ]

Simplify this expression to get the derivative of ( y ) with respect to ( x ).

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