Find the Derivative - d/dx x^5 natural log of 3x

Problem

d()/d(x)*x^5*ln(3*x)

Solution

1. Identify the rule needed for the derivative, which is the product rule d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x) where u=x^5 and v=ln(3*x)

2. Differentiate the first part u=x^5 using the power rule.

d(x^5)/d(x)=5*x^4

3. Differentiate the second part v=ln(3*x) using the chain rule.

d(ln(3*x))/d(x)=1/(3*x)⋅3

d(ln(3*x))/d(x)=1/x

4. Apply the product rule by combining the parts.

d()/d(x)*x^5*ln(3*x)=x^5⋅1/x+ln(3*x)⋅5*x^4

5. Simplify the expression by reducing the first term and rearranging the second term.

d()/d(x)*x^5*ln(3*x)=x^4+5*x^4*ln(3*x)

6. Factor out the common term x^4 to reach the final form.

d()/d(x)*x^5*ln(3*x)=x^4*(1+5*ln(3*x))

Final Answer

d()/d(x)*x^5*ln(3*x)=x^4*(1+5*ln(3*x))

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