Find the Derivative - d/d@VAR f(x)=x^5 natural log of x

Problem

d()/d(x)*x5*ln(x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=x5 and v(x)=ln(x)

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

3. Differentiate the individual components: d(x5)/d(x)=5*x4 and d(ln(x))/d(x)=1/x

4. Substitute these derivatives back into the product rule formula:

x5⋅1/x+ln(x)⋅5*x4

5. Simplify the expression by reducing the powers of x and rearranging terms:

x4+5*x4*ln(x)

6. Factor out the common term x4 to reach the final form.

Final Answer

(d(x5)*ln(x))/d(x)=x4*(1+5*ln(x))

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