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

Problem

d()/d(x)ln(x)/(x^4)

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two terms, ln(x) and x^4 use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Assign the variables for the quotient rule where u=ln(x) and v=x^4

3. Differentiate the individual components.

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

d(x^4)/d(x)=4*x^3

4. Substitute these derivatives into the quotient rule formula.

d(y)/d(x)=(x^4⋅1/x−ln(x)⋅4*x^3)/((x^4)^2)

5. Simplify the numerator by multiplying the terms.

d(y)/d(x)=(x^3−4*x^3*ln(x))/(x^8)

6. Factor out the common term x^3 from the numerator.

d(y)/d(x)=(x^3*(1−4*ln(x)))/(x^8)

7. Reduce the fraction by dividing both the numerator and denominator by x^3

d(y)/d(x)=(1−4*ln(x))/(x^5)

Final Answer

d()/d(x)ln(x)/(x^4)=(1−4*ln(x))/(x^5)

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