How do you differentiate #f(x)=(x^2+1)^3 # using the chain rule?
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To differentiate ( f(x) = (x^2 + 1)^3 ) using the chain rule:

Identify the outer function ( u ) and the inner function ( v ). Let ( u = (v)^3 ) and ( v = x^2 + 1 ).

Differentiate the outer function with respect to the inner function. ( \frac{du}{dv} = 3v^2 ).

Differentiate the inner function with respect to ( x ). ( \frac{dv}{dx} = 2x ).

Apply the chain rule: ( \frac{df}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = 3(v^2) \cdot (2x) ).

Substitute the expressions for ( u ) and ( v ): ( \frac{df}{dx} = 3(x^2 + 1)^2 \cdot (2x) ).
Therefore, ( \frac{df}{dx} = 6x(x^2 + 1)^2 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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