Find the Derivative - d/dx f(x)=5x^4 natural log of x

Problem

d()/d(x)*5*x4*ln(x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=5*x4 and v(x)=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 by applying the power rule to 5*x4

(d(5)*x4)/d(x)=20*x3

4. Differentiate the second part by applying the derivative rule for the natural logarithm.

d(ln(x))/d(x)=1/x

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

d()/d(x)*5*x4*ln(x)=5*x4⋅1/x+ln(x)⋅20*x3

6. Simplify the expression by multiplying the terms.

d()/d(x)*5*x4*ln(x)=5*x3+20*x3*ln(x)

7. Factor out the greatest common factor, 5*x3 to reach the final form.

d()/d(x)*5*x4*ln(x)=5*x3*(1+4*ln(x))

Final Answer

d()/d(x)*5*x4*ln(x)=5*x3*(1+4*ln(x))

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