# How do you differentiate #y= root3 (5x)^x#?

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To differentiate ( y = \sqrt{3}(5x)^x ), you can use the product rule combined with the chain rule.

- First, differentiate the term (\sqrt{3}) which is a constant and its derivative is 0.
- Then, differentiate the term ( (5x)^x ) using the chain rule.

Applying the chain rule, you get:

[ \frac{dy}{dx} = \sqrt{3} \cdot \frac{d}{dx} \left( (5x)^x \right) ]

Using the chain rule again, the derivative of ((5x)^x) is:

[ \frac{d}{dx} \left( (5x)^x \right) = (5x)^x \cdot \left( \ln{(5x)} + 1 \right) ]

So, putting it all together:

[ \frac{dy}{dx} = \sqrt{3} \cdot (5x)^x \cdot \left( \ln{(5x)} + 1 \right) ]

That's the derivative of ( y = \sqrt{3}(5x)^x ).

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