Find the Derivative - d/dx f(x)=3x^4 natural log of x

Problem

d()/d(x)*(3*x4*ln(x))

Solution

1. Identify the rule needed for the expression, which is the product rule for two functions u=3*x4 and v=ln(x)

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

3. Differentiate the individual components: (d(3)*x4)/d(x)=12*x3 and d(ln(x))/d(x)=1/x

4. Substitute these derivatives back into the product rule expression: 3*x(1/x)4+ln(x)*(12*x3)

5. Simplify the terms by multiplying and combining like powers of x

6. Factor out the greatest common factor, 3*x3 to reach the final form.

Final Answer

d()/d(x)*(3*x4*ln(x))=3*x3*(1+4*ln(x))

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