Find the Derivative - d/dx x^7+8x^7 natural log of x

Problem

d()/d(x)*(x7+8*x7*ln(x))

Solution

1. Identify the expression as a sum of two terms and apply the sum rule for derivatives.

2. Differentiate the first term, x7 using the power rule.

d(x7)/d(x)=7*x6

3. Apply the product rule to the second term, 8*x7*ln(x) which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

4. Calculate the derivative of the second term by setting u=8*x7 and v=ln(x)

(d(8)*x7*ln(x))/d(x)=8*x7⋅1/x+ln(x)⋅56*x6

5. Simplify the resulting expression from the product rule.

8*x6+56*x6*ln(x)

6. Combine the results from the first and second terms.

7*x6+8*x6+56*x6*ln(x)

7. Simplify the final expression by adding like terms.

15*x6+56*x6*ln(x)

Final Answer

d()/d(x)*(x7+8*x7*ln(x))=15*x6+56*x6*ln(x)

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