# How do you find the derivative of #F(x)=cose^tdt# from #[pi, lnx]#?

The Fundamental Theorem of Calculus, part 1, tells us that:

So:

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To find the derivative of (F(x) = \int_{\pi}^{\ln(x)} \cos(e^t) , dt) with respect to (x), we can apply the Fundamental Theorem of Calculus and the Chain Rule.

[\frac{d}{dx} F(x) = \frac{d}{dx} \left( \int_{\pi}^{\ln(x)} \cos(e^t) , dt \right)]

By the Fundamental Theorem of Calculus, if (F(x) = \int_{g(x)}^{h(x)} f(t) , dt), then (F'(x) = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x)).

In our case, (f(t) = \cos(e^t)), (g(x) = \pi), and (h(x) = \ln(x)). So, we substitute these values into the formula:

[F'(x) = \cos(e^{\ln(x)}) \cdot \frac{d}{dx} \ln(x) - \cos(e^{\pi}) \cdot \frac{d}{dx} \pi]

[= \cos(x) \cdot \frac{1}{x} - \cos(e^{\pi}) \cdot 0]

[= \frac{\cos(x)}{x}]

Therefore, the derivative of (F(x)) is (\frac{\cos(x)}{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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