# How do you find the derivative of #g(x)=(2x^2+x+1)^-3#?

We use the chain rule, which states that,

And so,

I'd clean this up and rearrange it into:

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To find the derivative of ( g(x) = (2x^2 + x + 1)^{-3} ), you can use the chain rule.

- Start by applying the chain rule, which states that if ( u ) is a function of ( x ) and ( f(u) ) is a function of ( u ), then the derivative of ( f(u) ) with respect to ( x ) is ( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} ).
- Let ( u = 2x^2 + x + 1 ).
- Find the derivative of ( u ) with respect to ( x ), which is ( \frac{du}{dx} = 4x + 1 ).
- Find the derivative of ( g(u) = u^{-3} ) with respect to ( u ), which is ( \frac{dg}{du} = -3u^{-4} ).
- Multiply the derivatives from steps 3 and 4 together to get the derivative of the original function.
- Substitute ( u = 2x^2 + x + 1 ) back into the result.

Therefore, the derivative of ( g(x) ) with respect to ( x ) is ( g'(x) = -3(2x^2 + x + 1)^{-4} \cdot (4x + 1) ).

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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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