# How do you use the Fundamental Theorem of Calculus to find the derivative of #f(x)=int_1^(x)sqrt(e^t+sin(t))dt# ?

Fundamental Theorem of Calculus

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To find the derivative of ( f(x) = \int_1^x \sqrt{e^t + \sin(t)} , dt ) using the Fundamental Theorem of Calculus, you differentiate the integral function with respect to ( x ). The Fundamental Theorem of Calculus states that if ( F(x) ) is an antiderivative of ( f(x) ), then ( \frac{d}{dx} \left( \int_a^x f(t) , dt \right) = f(x) ). Therefore, to differentiate ( f(x) ), we find the antiderivative of the integrand and then differentiate it with respect to ( x ). The integrand is ( \sqrt{e^t + \sin(t)} ). To find its antiderivative, we'll denote ( u = e^t + \sin(t) ), and differentiate ( u ) with respect to ( t ) to find ( du ). Then integrate ( \sqrt{u} ) with respect to ( t ), substitute back ( u ) in terms of ( x ), and finally differentiate the result 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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