How do you find the derivative of #F(x) = int sqrt(1+sec(3t)) dt#?

Answer 1

Without knowing the limits of integration all we can say if the following.

For #F(x) = int_(f(x))^g(x) sqrt(1+sec(3t)) dt#

The Fundamental Theorem of Calculus gives us

#F'(x) = sqrt(1+sec(3g(x))) g'(x) - sqrt(1+sec(3f(x))) f'(x)#

So

Example 1 For #F(x) = int_(-pi/8)^x sqrt(1+sec(3t)) dt#

The Fundamental Theorem of Calculus gives us

#F'(x) = sqrt(1+sec(3x)#

Example 2

For #F(x) = int_0^x^2 sqrt(1+sec(3t)) dt#

The Fundamental Theorem of Calculus gives us

#F'(x) = 2xsqrt(1+sec(3x^2))#

Example 3

For #F(x) = int_x^x^2 sqrt(1+sec(3t)) dt#

The Fundamental Theorem of Calculus gives us

#F'(x) = 2xsqrt(1+sec(3x^2)) - sqrt(1+sec(3x))#
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Answer 2

To find the derivative of ( F(x) = \int \sqrt{1 + \sec(3t)} , dt ), we can use the chain rule for differentiation. The chain rule states that if ( F(x) = \int g(u) , du ), then ( F'(x) = g(u) \cdot u' ), where ( u ) is a function of ( x ).

In this case, let's let ( u = 3t ). Then ( du = 3 , dt ). Now we can rewrite the integral in terms of ( u ):

[ F(x) = \int \sqrt{1 + \sec(u)} , \frac{du}{3} ]

Now differentiate ( F(x) ) with respect to ( x ) using the chain rule:

[ F'(x) = \frac{1}{3} \cdot \frac{d}{dx} \left( \sqrt{1 + \sec(u)} \right) ]

To find ( \frac{d}{dx} \left( \sqrt{1 + \sec(u)} \right) ), we first find its derivative with respect to ( u ) and then multiply by ( \frac{du}{dx} ), which is ( 3 ):

[ \frac{d}{du} \left( \sqrt{1 + \sec(u)} \right) = \frac{1}{2} \frac{1}{\sqrt{1 + \sec(u)}} \cdot \sec(u) \tan(u) ]

Now multiply by ( \frac{du}{dx} = 3 ) to get the final derivative:

[ F'(x) = \frac{1}{6} \cdot \frac{\sec(u) \tan(u)}{\sqrt{1 + \sec(u)}} \cdot 3 ]

Simplifying, we have:

[ F'(x) = \frac{\sec(u) \tan(u)}{2\sqrt{1 + \sec(u)}} ]

Finally, substitute back ( u = 3t ) to get the derivative in terms of ( x ):

[ F'(x) = \frac{\sec(3t) \tan(3t)}{2\sqrt{1 + \sec(3t)}} ]

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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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