How do you differentiate # g(x) = 3xsin^2(4x) + secx #?

Answer 1

Make use of the Chain Rule and Product Rule. #(dg)/dx = 3sin^2(4x) + 24xsin(4x)cos(4x)+tan(x)sec(x)#

Our first term will have to have the product rule applied. The product rule states that given #f(x)=g(x)h(x), f'(x) = g'(x)h(x) + g(x)h'(x)#

Thus our first term...

#d/dx (3xsin^2(4x)) = 3sin^2(4x) + 3x d/dx (sin^2 (4x))#
The derivative for this second term will require use of the chain rule, and of substitution. If we declare #u(x) = sin(4x)#, then we have #(du)/dx = 4 cos(4x) sin^2(4x) = u^2#, meaning #d/dx u^2 = d/du (u^2) * (du)/dx = 2u*(du)/dx. = 2 sin(4x) * 4cos(4x) = 8sin(4x)cos(4x)#

This means:

#d/dx (3xsin^2(4x)) = 3sin^2(4x) + 3x(8 sin(4x)cos(4x)) = 3sin^2(4x) + 24xsin(4x)cos(4x)#
Meanwhile, the derivative of #sec(x)# can be found using the quotient rule and the definition of the secant.
#sec(x) = 1/cos(x) -> d/dx (1/cos x) = ((0*cosx) - (-sin x))/(cos^2x) = sin x / (cos^2x) = tan(x)* 1/cos(x) = tanx secx#

Thus we have...

#(dg)/dx = 3sin^2(4x) + 24xsin(4x)cos(4x)+tan(x)sec(x)#
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Answer 2

To differentiate the function g(x) = 3xsin^2(4x) + sec(x):

  1. Differentiate each term separately using the rules of differentiation.
  2. 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.
  3. 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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Answer 3

To differentiate the function ( g(x) = 3x\sin^2(4x) + \sec(x) ), you can use the following steps:

  1. Differentiate each term of the function separately using the derivative rules.
  2. Apply the chain rule where necessary for functions like ( \sin^2(4x) ) and ( \sec(x) ).

Let's differentiate each term step by step:

  1. 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 ]

  2. 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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Answer from HIX Tutor

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.

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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