Find the Derivative - d/dx x^2arctan(x)

Problem

d()/d(x)*x2*arctan(x)

Solution

1. Identify the rule needed for the expression, which is the product of two functions: u=x2 and v=arctan(x)

2. Apply the product rule, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part of the product: d(x2)/d(x)=2*x

4. Differentiate the second part of the product: d(arctan(x))/d(x)=1/(1+x2)

5. Substitute these derivatives back into the product rule formula.

x2d(arctan(x))/d(x)+arctan(x)d(x2)/d(x)

6. Simplify the resulting expression by performing the multiplication.

x21/(1+x2)+arctan(x)*(2*x)

(x2)/(1+x2)+2*x*arctan(x)

Final Answer

(d(x2)*arctan(x))/d(x)=(x2)/(1+x2)+2*x*arctan(x)

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