# How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function #y = int (6+v^8)^3 dv# from 1 to cos(x)?

The answer is

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To find the derivative of the function ( y = \int_{1}^{\cos(x)} (6+v^8)^3 , dv ), you can use Part 1 of the Fundamental Theorem of Calculus, which states that if ( F(x) ) is the integral of ( f(t) ) from a constant ( a ) to ( x ), then ( F'(x) = f(x) ).

Apply this theorem to the given function by first finding the antiderivative of ( (6+v^8)^3 ) with respect to ( v ). Let ( F(v) ) represent this antiderivative. Then, ( F'(\cos(x)) ) will be the derivative of the function with respect to ( x ).

Finally, find ( F'(\cos(x)) ) by differentiating ( F(v) ) with respect to ( v ) and then substituting ( \cos(x) ) for ( v ) in the resulting expression. This gives you the derivative of the function with respect to ( x ).

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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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