# How do you differentiate #f(x)=x^7sqrt(4x^2+7)# using the product rule?

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Now that we have all the data required, we can apply the product rule.

There's more simplification possible, but I'll let you handle the algebra.

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To differentiate ( f(x) = x^7 \sqrt{4x^2 + 7} ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( u(x) = x^7 ) and ( v(x) = \sqrt{4x^2 + 7} ).
- Apply the product rule formula: ( (uv)' = u'v + uv' ).
- Find the derivatives of ( u(x) ) and ( v(x) ) individually:
- ( u'(x) = 7x^6 ) (using the power rule)
- ( v'(x) = \frac{1}{2\sqrt{4x^2 + 7}} \cdot 8x ) (using the chain rule)

- Substitute these derivatives into the product rule formula:
- ( f'(x) = (7x^6)(\sqrt{4x^2 + 7}) + (x^7)\left(\frac{1}{2\sqrt{4x^2 + 7}} \cdot 8x\right) )

- Simplify the expression:
- ( f'(x) = 7x^6 \sqrt{4x^2 + 7} + 4x^8 )

So, the derivative of ( f(x) = x^7 \sqrt{4x^2 + 7} ) using the product rule is ( f'(x) = 7x^6 \sqrt{4x^2 + 7} + 4x^8 ).

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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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