# How do you differentiate #y=(x^2lnx)^4#?

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To differentiate ( y = (x^2 \ln x)^4 ), you would use the chain rule. Here's the step-by-step process:

- Let ( u = x^2 \ln x ).
- Differentiate ( u ) with respect to ( x ) to find ( \frac{du}{dx} ).
- Apply the power rule to ( u^4 ) to find ( \frac{dy}{du} ).
- Multiply ( \frac{du}{dx} ) and ( \frac{dy}{du} ) to find ( \frac{dy}{dx} ), using the chain rule.

Here's the breakdown:

- ( u = x^2 \ln x )
- ( \frac{du}{dx} = 2x \ln x + x )

- ( \frac{dy}{du} = 4u^3 )
- ( \frac{dy}{du} = 4(x^2 \ln x)^3 )

- ( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} )
- ( \frac{dy}{dx} = 4(x^2 \ln x)^3 \times (2x \ln x + x) )

So, the derivative of ( y = (x^2 \ln x)^4 ) with respect to ( x ) is:

[ \frac{dy}{dx} = 4(x^2 \ln x)^3 \times (2x \ln x + x) ]

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