# How do you use the chain rule to differentiate #(sqrtx)^10#?

Using the chain rule, we get

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To differentiate ((\sqrt{x})^{10}) using the chain rule, follow these steps:

- Identify the outer function, which is (u = x^{10}).
- Identify the inner function, which is (v = \sqrt{x}).
- Find the derivative of the outer function with respect to the inner function: (\frac{du}{dv}).
- Find the derivative of the inner function with respect to (x): (\frac{dv}{dx}).
- Apply the chain rule formula: (\frac{du}{dx} = \frac{du}{dv} \times \frac{dv}{dx}).

By differentiating (u = x^{10}) with respect to (v = \sqrt{x}), we get (\frac{du}{dv} = 10v^9). By differentiating (v = \sqrt{x}) with respect to (x), we get (\frac{dv}{dx} = \frac{1}{2\sqrt{x}}).

Now, apply the chain rule:

[\frac{du}{dx} = \frac{du}{dv} \times \frac{dv}{dx} = 10v^9 \times \frac{1}{2\sqrt{x}} = \frac{5v^9}{\sqrt{x}}]

Replace (v) with (\sqrt{x}) to get the final answer:

[\frac{du}{dx} = \frac{5(\sqrt{x})^9}{\sqrt{x}} = 5x^{\frac{9}{2}}]

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To differentiate ((\sqrt{x})^{10}) using the chain rule, first express the function as ((x^{1/2})^{10}). Then, apply the chain rule, which states that if (u = f(g(x))), then (u' = f'(g(x)) \cdot g'(x)). In this case, (f(u) = u^{10}) and (g(x) = x^{1/2}).

Now, find the derivatives of (f(u)) and (g(x)). The derivative of (f(u) = u^{10}) with respect to (u) is (f'(u) = 10u^9). The derivative of (g(x) = x^{1/2}) with respect to (x) is (g'(x) = \frac{1}{2}x^{-1/2}).

Finally, apply the chain rule formula: [ \frac{d}{dx} \left((\sqrt{x})^{10}\right) = f'(g(x)) \cdot g'(x) = 10 \left(x^{1/2}\right)^9 \cdot \frac{1}{2}x^{-1/2} ]

Simplify this expression to get the final result.

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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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