Find the Derivative - d/d@VAR f(x)=x^6 natural log of 7x

Problem

d()/d(x)*x6*ln(7*x)

Solution

1. Identify the function as a product of two terms, u=x6 and v=ln(7*x) which requires the product rule d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first term u=x6 using the power rule.

d(x6)/d(x)=6*x5

3. Differentiate the second term v=ln(7*x) using the chain rule.

d(ln(7*x))/d(x)=1/(7*x)⋅7

d(ln(7*x))/d(x)=1/x

4. Apply the product rule formula by substituting the derivatives found in the previous steps.

d()/d(x)*x6*ln(7*x)=x6⋅1/x+ln(7*x)⋅6*x5

5. Simplify the expression by performing the multiplication and factoring out the greatest common factor.

d()/d(x)*x6*ln(7*x)=x5+6*x5*ln(7*x)

d()/d(x)*x6*ln(7*x)=x5*(1+6*ln(7*x))

Final Answer

d()/d(x)*x6*ln(7*x)=x5*(1+6*ln(7*x))

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