Find the Derivative - d/dx y=x^5 natural log of x-1/3x^3

Problem

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

4. Differentiate the second part of the product rule:

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

5. Apply the power rule to the second term of the original expression, −1/3*x^3

d()/d(x)*(−1/3*x^3)=−1/3⋅3*x^2=−x^2

6. Combine all the resulting terms to find the final derivative.

d(y)/d(x)=x^4+5*x^4*ln(x)−x^2

Final Answer

d()/d(x)*(x^5*ln(x)−1/3*x^3)=x^4+5*x^4*ln(x)−x^2

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