# How do you find the derivative of #arcsin e^x#?

There are two methods:

Using the pre-memorized arcsine derivative:

You may already know that the derivative of arcsine is:

Without knowing the arcsine derivative:

Let

Thus:

Differentiate both sides (the chain rule will be used on the left-hand side!):

We know that

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To find the derivative of arcsin(e^x), you can use the chain rule. The derivative of arcsin(u) with respect to u is 1/sqrt(1-u^2), and the derivative of e^x with respect to x is e^x. Applying the chain rule, the derivative of arcsin(e^x) with respect to x is:

d(arcsin(e^x))/dx = (1/sqrt(1-(e^x)^2)) * (d(e^x)/dx) = (1/sqrt(1-e^(2x))) * 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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