Find the differential of the function.

(a) y=x^2 sin2x

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I think you mean differential of y = x² sinx.

First we find the derivative of sin2x.

Using double angle formula sin2x= 2sinx.cosx, we get

d(sin2x)/dx = d(2sinx cosx)/dx

= 2d(sinx.cosx)/dx

now using product rule i.e. `(d(uv))/dx=u(dv)/dx + v(du)/dx`, we get

d(sin2x)/dx = 2(sinx . d(cosx)/dx + cosx.d(sinx)/dx)

= 2(sinx.(-sinx) + cosx.(cosx))

= 2(cos²x - sin²x)

We know, cos(2x) = cos²x - sin²x -------- double angle formula.

therefore, d(sin2x)/dx = 2cos2x ------------ [1]

Now y = x²sin2x

therefore, dy/dx = d(x²sin2x)/dx

Again using product rule,

= (x²)(d(sin2x)/dx) + (sin2x)(d(x^2)/dx)

from [1] = (x²)(2cos2x) + (sin2x)(2x)

=2x²cos2x + 2xsin2x

**dy/dx = 2x²cos2x + 2xsin2x**

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