Find the Derivative - d/dx 5x natural log of x

Problem

d()/d(x)*5*x*ln(x)

Solution

1. Identify the expression as a product of two functions, 5*x and ln(x) which requires the use of the product rule.

2. Apply the product rule formula, which states that d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

3. Assign the variables such that u=5*x and v=ln(x)

4. Differentiate each part individually to find (d(5)*x)/d(x)=5 and d(ln(x))/d(x)=1/x

5. Substitute these components back into the product rule formula:

5*x⋅1/x+ln(x)⋅5

6. Simplify the resulting expression by canceling x in the first term and rearranging the second term.

5+5*ln(x)

Final Answer

(d(5)*x*ln(x))/d(x)=5+5*ln(x)

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