Find the Derivative - d/dx y=( natural log of x)/(x^12)

Problem

d()/d(x)ln(x)/(x12)

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two functions, u=ln(x) and v=x12 use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Differentiate the numerator and the denominator separately.

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

d(x12)/d(x)=12*x11

3. Apply the quotient rule formula by substituting the functions and their derivatives.

d()/d(x)ln(x)/(x12)=(x12⋅1/x−ln(x)⋅12*x11)/((x12)2)

4. Simplify the terms in the numerator and the power in the denominator.

d()/d(x)ln(x)/(x12)=(x11−12*x11*ln(x))/(x24)

5. Factor out the common term x11 from the numerator to further simplify the expression.

d()/d(x)ln(x)/(x12)=(x11*(1−12*ln(x)))/(x24)

6. Reduce the fraction by dividing both the numerator and denominator by x11

d()/d(x)ln(x)/(x12)=(1−12*ln(x))/(x13)

Final Answer

d()/d(x)ln(x)/(x12)=(1−12*ln(x))/(x13)

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