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

Problem

d()/d(x)*x^8*ln(x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x^8 and ln(x) the product rule must be applied: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions to variables. Let u=x^8 and v=ln(x)

3. Differentiate each part separately. The derivative of u is d(x^8)/d(x)=8*x^7 using the power rule. The derivative of v is d(ln(x))/d(x)=1/x

4. Apply the product rule formula by substituting the functions and their derivatives.

(d(x^8)*ln(x))/d(x)=x^8⋅1/x+ln(x)⋅8*x^7

5. Simplify the terms. Multiplying x^8 by 1/x results in x^7

(d(x^8)*ln(x))/d(x)=x^7+8*x^7*ln(x)

6. Factor out the greatest common factor, which is x^7 to reach the final form.

(d(x^8)*ln(x))/d(x)=x^7*(1+8*ln(x))

Final Answer

(d(x^8)*ln(x))/d(x)=x^7*(1+8*ln(x))

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