Find the Derivative - d/dx y=x^7 natural log of x-1/3x^3

Problem

d()/d(x)*(x^7*ln(x)−1/3*x^3)

Solution

1. Identify the structure of the expression as a difference of two terms, allowing the use of the sum/difference rule for derivatives.

2. Apply the product rule to the first term x^7*ln(x) where the rule is (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part of the product rule: x^7⋅d(ln(x))/d(x)=x^7⋅1/x=x^6

4. Differentiate the second part of the product rule: ln(x)⋅d(x^7)/d(x)=ln(x)⋅7*x^6

5. Apply the power rule to the second term of the original expression: d()/d(x)1/3*x^3=1/3⋅3*x^2=x^2

6. Combine all the results and simplify the expression by grouping the terms from the product rule and subtracting the derivative of the second term.

7. Factor out the common term x^2 to provide a simplified final form.

Final Answer

d()/d(x)*(x^7*ln(x)−1/3*x^3)=x^6+7*x^6*ln(x)−x^2

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