How do you use part 1 of Fundamental Theorem of Calculus to find the derivative of the function #y=int (6+v^2)^10 dv# from sinx to cosx?

Answer 1
#[-(6+cos^2x)^10 sinx - (6+sin^2x)^10 cosx]#
If it is F(v) = #int_sinx^cosx (6+v^2)#dv, it can be written as a sum of two integrals= #int_sinx^0 (6+v^2)dv# +#int_0^cosx (6+v^2)dv#
=#int_0^cosx (6+v^2)dv-int_0^sinx (6+v^2)dv#
F'(v) =#d/dx int_0^cosx (6+v^2)dv -d/dx int_o^sinx (6+v^2)dv#
Now applying the Fundamental Theorem of Calculus, the derivative would be #[-(6+cos^2x)^10 sinx - (6+sin^2x)^10 cosx]#
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Answer 2

To find the derivative of the function ( y = \int_{\sin x}^{\cos x} (6+v^2)^{10} , dv ) using Part 1 of the Fundamental Theorem of Calculus, we first need to evaluate the integrand at the upper limit (cos x) and the lower limit (sin x). Then, we differentiate the result with respect to x. The derivative of the function with respect to x is given by:

[ \frac{d}{dx} \left[ \int_{\sin x}^{\cos x} (6+v^2)^{10} , dv \right] = (6 + \cos^2 x)^{10} \cdot \frac{d}{dx} \cos x - (6 + \sin^2 x)^{10} \cdot \frac{d}{dx} \sin x ]

[ = (6 + \cos^2 x)^{10} \cdot (-\sin x) - (6 + \sin^2 x)^{10} \cdot \cos x ]

[ = -\sin x \cdot (6 + \cos^2 x)^{10} - \cos x \cdot (6 + \sin^2 x)^{10} ]

This is the derivative of the given function with respect to 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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