Find the derivative of the function. `f(x) = (1 - 4x - x^2)( x^2 - 4)`  

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lemjay | High School Teacher | (Level 3) Senior Educator

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To take the derivative of the given function, apply the power rule which is `(u*v)'=v*u'+u*v'` .

So let,

`u=1-4x-x^2`        and        `v=x^2-4`

Then, take the derivative of u and v to get u' and v'.

`u'=-4-2x`             and        `v'=2x`         

And, plug-in u,v, u' and v' to the formula of product rule.

`f'(x)=(x^2-4)(-4-2x) + (1-4x-x^2)(2x)`

Then, expand.

`f'(x)= -4x^2 -2x^3+16+8x+2x - 8x^2-2x^3`

Combine like terms.

`f'(x) = -4x^3-12x^2+10x+16`     

Hence, the derivative of the given function is `f'(x)=-4x^3-12x^2+10x+16` .