Find the Derivative - d/dx y=x^2 natural log of 6x

Problem

d()/d(x)*x2*ln(6*x)

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x2 and v=ln(6*x)

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

3. Differentiate the first part u=x2 to get d(u)/d(x)=2*x

4. Differentiate the second part v=ln(6*x) using the chain rule to get d(v)/d(x)=1/(6*x)⋅6=1/x

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

x2⋅1/x+ln(6*x)⋅2*x

6. Simplify the resulting expression by canceling x in the first term and rearranging the second term.

x+2*x*ln(6*x)

7. Factor out the common term x to reach the final form.

x*(1+2*ln(6*x))

Final Answer

d()/d(x)*x2*ln(6*x)=x*(1+2*ln(6*x))

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