# How do you differentiate #f(x)=sin(6x+5x^2+1)# using the chain rule?

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To differentiate ( f(x) = \sin(6x + 5x^2 + 1) ) using the chain rule:

- Identify the outer function ( \sin(u) ) and the inner function ( u = 6x + 5x^2 + 1 ).
- Take the derivative of the outer function with respect to its input: ( \frac{d}{du} \sin(u) = \cos(u) ).
- Take the derivative of the inner function with respect to ( x ): ( \frac{du}{dx} = 12x + 5 ).
- Multiply the derivatives found in steps 2 and 3: ( \cos(u) \cdot (12x + 5) ).

Therefore, the derivative of ( f(x) = \sin(6x + 5x^2 + 1) ) with respect to ( x ) using the chain rule is ( f'(x) = (12x + 5) \cdot \cos(6x + 5x^2 + 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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