# How do you differentiate #y=(8x+sec(x))^7#?

Using the chain rule:

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To differentiate ( y = (8x + \sec(x))^7 ):

- Identify the inner function ( u(x) = 8x + \sec(x) ).
- Apply the chain rule: [ \frac{d}{dx} (u(x))^7 = 7(u(x))^{6} \cdot \frac{d}{dx} (u(x)) ]
- Compute the derivative of the inner function:
- ( \frac{d}{dx} (8x) = 8 )
- ( \frac{d}{dx} (\sec(x)) = \sec(x) \tan(x) )

- Plug the derivative and the original function into the chain rule formula: [ \frac{d}{dx} (8x + \sec(x))^7 = 7(8x + \sec(x))^6 \left(8 + \sec(x) \tan(x)\right) ]

So, the derivative of ( y = (8x + \sec(x))^7 ) with respect to ( x ) is ( \frac{d}{dx} (8x + \sec(x))^7 = 7(8x + \sec(x))^6 \left(8 + \sec(x) \tan(x)\right) ).

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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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