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

Problem

d()/d(x)*(x7*ln(x)−1/3*x3)

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 x7*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: x7⋅d(ln(x))/d(x)=x7⋅1/x=x6

4. Differentiate the second part of the product rule: ln(x)⋅d(x7)/d(x)=ln(x)⋅7*x6

5. Apply the power rule to the second term of the original expression: d()/d(x)1/3*x3=1/3⋅3*x2=x2

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 x2 to provide a simplified final form.

Final Answer

d()/d(x)*(x7*ln(x)−1/3*x3)=x6+7*x6*ln(x)−x2

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