Find the Derivative - d/dx y=x^3 natural log of x

Problem

d()/d(x)*x^3*ln(x)

Solution

1. Identify the rule needed for the expression, which is the product of two functions: u=x^3 and v=ln(x)

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

3. Differentiate the individual components: d(x^3)/d(x)=3*x^2 and d(ln(x))/d(x)=1/x

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

(d(x^3)*ln(x))/d(x)=x^3⋅1/x+ln(x)⋅3*x^2

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

(d(x^3)*ln(x))/d(x)=x^2+3*x^2*ln(x)

6. Factor out the common term x^2

(d(x^3)*ln(x))/d(x)=x^2*(1+3*ln(x))

Final Answer

(d(x^3)*ln(x))/d(x)=x^2*(1+3*ln(x))

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