Find the Derivative - d/dx 2x^4 natural log of x

Problem

d()/d(x)*2*x4*ln(x)

Solution

1. Identify the rule needed for the expression, which is the product of two functions: u=2*x4 and v=ln(x)

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

3. Differentiate the first part u=2*x4 using the power rule to get (d(2)*x4)/d(x)=8*x3

4. Differentiate the second part v=ln(x) to get d(ln(x))/d(x)=1/x

5. Substitute these derivatives back into the product rule formula.

2*x4⋅1/x+ln(x)⋅8*x3

6. Simplify the terms by dividing 2*x4 by x and rearranging the expression.

2*x3+8*x3*ln(x)

7. Factor out the common term 2*x3 to reach the final form.

2*x3*(1+4*ln(x))

Final Answer

(d(2)*x4*ln(x))/d(x)=2*x3*(1+4*ln(x))

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