# How do you find the derivative of function #y=(sqrtx) *(x-3)^2#?

In this way:

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To find the derivative of the function ( y = \sqrt{x} \times (x - 3)^2 ), you can use the product rule of differentiation. The product rule states that the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Applying the product rule, we get:

[ y' = \frac{d}{dx}[\sqrt{x}] \times (x - 3)^2 + \sqrt{x} \times \frac{d}{dx}[(x - 3)^2] ]

Now, find the derivatives of each term using the power rule and the chain rule:

[ \frac{d}{dx}[\sqrt{x}] = \frac{1}{2\sqrt{x}} ]

[ \frac{d}{dx}[(x - 3)^2] = 2(x - 3) ]

Substitute these derivatives back into the product rule equation:

[ y' = \frac{1}{2\sqrt{x}} \times (x - 3)^2 + \sqrt{x} \times 2(x - 3) ]

[ y' = \frac{(x - 3)^2}{2\sqrt{x}} + 2\sqrt{x}(x - 3) ]

[ y' = \frac{(x - 3)^2}{2\sqrt{x}} + 2x\sqrt{x} - 6\sqrt{x} ]

[ y' = \frac{(x - 3)^2 + 4x\sqrt{x} - 12\sqrt{x}}{2\sqrt{x}} ]

So, the derivative of the function ( y = \sqrt{x} \times (x - 3)^2 ) is ( \frac{(x - 3)^2 + 4x\sqrt{x} - 12\sqrt{x}}{2\sqrt{x}} ).

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