# How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function #y = int sin^3 t dt# from #[e^x, 0]#?

The theorem says:

Given the function:

than:

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To use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function ( y = \int_{e^x}^{0} \sin^3(t) , dt ), we first need to evaluate the integral and then differentiate the result with respect to ( x ). Let's denote the integral as a function of ( x ), say ( F(x) ). Then, according to the Fundamental Theorem of Calculus Part 1, the derivative of ( F(x) ) with respect to ( x ) is given by ( \frac{d}{dx} \int_{e^x}^{0} \sin^3(t) , dt = -\sin^3(e^x) \cdot \frac{d}{dx}(e^x) ).

Differentiating ( e^x ) with respect to ( x ) gives ( \frac{d}{dx}(e^x) = e^x ).

So, the derivative of the function ( y = \int_{e^x}^{0} \sin^3(t) , dt ) with respect to ( x ) is ( -\sin^3(e^x) \cdot e^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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