# What is the derivative of #(x^3 + 2)^2(x^5 + 4)^4#?

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To find the derivative of the given function, we can use the product rule and chain rule of differentiation.

Let ( u = (x^3 + 2)^2 ) and ( v = (x^5 + 4)^4 ).

Then, using the product rule, the derivative of the function ( f(x) = uv ) with respect to ( x ) is given by:

[ f'(x) = u'v + uv' ]

To find ( u' ) and ( v' ), we apply the chain rule:

[ u' = 2(x^3 + 2)(3x^2) ] [ v' = 4(x^5 + 4)^3(5x^4) ]

Substituting these into the product rule formula, we get:

[ f'(x) = (2(x^3 + 2)(3x^2))(x^5 + 4)^4 + (x^3 + 2)^2(4(x^5 + 4)^3(5x^4)) ]

Simplifying this expression yields the derivative of the given function:

[ f'(x) = 2(x^3 + 2)(3x^2)(x^5 + 4)^4 + 4(x^3 + 2)^2(x^5 + 4)^3(5x^4) ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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