How do you differentiate # g(x) = 3xsin^2(4x) + secx #?
Make use of the Chain Rule and Product Rule.
Thus our first term...
This means:
Thus we have...
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To differentiate the function g(x) = 3xsin^2(4x) + sec(x):
 Differentiate each term separately using the rules of differentiation.
 For the term 3xsin^2(4x):
 Apply the chain rule for the function sin^2(4x).
 The derivative of sin^2(4x) is 2sin(4x)cos(4x) * 4.
 Multiply the result by the derivative of the outer function, which is 3x.
 For the term sec(x):
 The derivative of sec(x) is sec(x)tan(x).
Therefore, the derivative of g(x) with respect to x is:
g'(x) = 3(sin^2(4x) + 8xsin(4x)cos(4x)) + sec(x)tan(x)
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To differentiate the function ( g(x) = 3x\sin^2(4x) + \sec(x) ), you can use the following steps:
 Differentiate each term of the function separately using the derivative rules.
 Apply the chain rule where necessary for functions like ( \sin^2(4x) ) and ( \sec(x) ).
Let's differentiate each term step by step:

Differentiate ( 3x\sin^2(4x) ): [ \frac{d}{dx} (3x\sin^2(4x)) = 3\sin^2(4x) + 3x \cdot 2\sin(4x) \cos(4x) \cdot 4 ]

Differentiate ( \sec(x) ): [ \frac{d}{dx} (\sec(x)) = \sec(x) \tan(x) ]
Combining the results from step 1 and step 2, the derivative of ( g(x) ) is: [ g'(x) = 3\sin^2(4x) + 24x\sin(4x)\cos(4x) + \sec(x)\tan(x) ]
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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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