# How do you differentiate #y=(sin^-1(5x^2))^3#?

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To differentiate ( y = (\sin^{-1}(5x^2))^3 ), you would use the chain rule and the power rule. The chain rule states that if you have a function within another function, you differentiate the outer function first and then multiply by the derivative of the inner function.

The derivative of (\sin^{-1}(u)) with respect to (u) is (\frac{1}{\sqrt{1 - u^2}}). So, when differentiating (y) with respect to (x), you would first differentiate the outer function ((\sin^{-1}(5x^2))^3), then multiply by the derivative of the inner function (5x^2).

The derivative of ((\sin^{-1}(5x^2))^3) with respect to (\sin^{-1}(5x^2)) is (3(\sin^{-1}(5x^2))^2), and the derivative of (5x^2) with respect to (x) is (10x).

Therefore, when you multiply these together, you get (3(\sin^{-1}(5x^2))^2 \cdot 10x), which simplifies to (30x(\sin^{-1}(5x^2))^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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