# How do you differentiate #y=7sin^-1(10x)#?

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To differentiate (y = 7\sin^{-1}(10x)), we'll use the chain rule. The derivative of (y = \sin^{-1}(x)) with respect to (x) is (\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}). Applying the chain rule to our function:

- Let (u = 10x), so (y = 7\sin^{-1}(u)).
- Differentiate (y) with respect to (u): (\frac{dy}{du} = \frac{7}{\sqrt{1-u^2}}).
- Differentiate (u = 10x) with respect to (x): (\frac{du}{dx} = 10).

Now, multiply the derivatives:

[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{7}{\sqrt{1-(10x)^2}} \cdot 10 = \frac{70}{\sqrt{1-100x^2}}. ]

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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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