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How do you calculate the derivative of #int5(sin(t))^5 dt# from #[e^x,5]#?

Answer 1

William Abdalla

# 5 d/dx int_(e^x)^5 sin^5 t dt = = -5 e^x sin^5 (e^x) #

# 5d/dx int_(e^x)^5 sin^5 t dt = -5 d/dxint_5^(e^x) sin^5 t dt # # = -5 sin^5 (e^x) d/dx(e^x) = -5 e^x sin^5 (e^x) #

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Answer 2

Emma Aldridge

To calculate the derivative of the integral (\int_{e^x}^5 (\sin(t))^5 , dt), you can use the Fundamental Theorem of Calculus and the Chain Rule. First, apply the Fundamental Theorem of Calculus to find the antiderivative of the integrand. Then, differentiate this antiderivative with respect to (x) using the Chain Rule. The result is:

[\frac{d}{dx}\left( \int_{e^x}^5 (\sin(t))^5 , dt \right) = -\sin((e^x))^5 \cdot \frac{d}{dx}(e^x)]

Simplify this expression to obtain the final derivative.

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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.