Find the Derivative - d/d@VAR f(x)=x^2 natural log of x^2

Problem

d()/d(x)*(x2*ln(x2))

Solution

1. Identify the function as a product of two terms, u=x2 and v=ln(x2) which requires the product rule d()/d(x)*(u*v)=ud(v)/d(x)+vd(u)/d(x)

2. Simplify the logarithmic term using the power rule for logarithms, ln(ab)=b*ln(a) to make differentiation easier.

ln(x2)=2*ln(x)

3. Rewrite the original function using the simplified logarithm.

ƒ(x)=2*x2*ln(x)

4. Apply the product rule to the rewritten function.

d(ƒ)/d(x)=(2*x2)d(ln(x))/d(x)+ln(x)(d(2)*x2)/d(x)

5. Differentiate the individual components.

d(ln(x))/d(x)=1/x

(d(2)*x2)/d(x)=4*x

6. Substitute these derivatives back into the product rule expression.

d(ƒ)/d(x)=2*x(1/x)2+ln(x)*(4*x)

7. Simplify the resulting expression by performing the multiplication and factoring.

d(ƒ)/d(x)=2*x+4*x*ln(x)

d(ƒ)/d(x)=2*x*(1+2*ln(x))

Final Answer

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

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