Find the Derivative - d/dx natural log of (e^(x^2)(3x-2)^7)/(7x^9)

Problem

d()/d(x)*ln((e(x2)*(3*x−2)7)/(7*x9))

Solution

1. Apply logarithm properties to expand the expression before differentiating. Use the rules ln(a/b)=ln(a)−ln(b) and ln(a*b)=ln(a)+ln(b)

ln((e(x2)*(3*x−2)7)/(7*x9))=ln(e(x2))+ln((3*x−2)7)−ln(7*x9)

2. Simplify the terms using the power rule ln(an)=n*ln(a) and the identity ln(eu)=u

ln(e(x2))+ln((3*x−2)7)−ln(7*x9)=x2+7*ln(3*x−2)−(ln(7)+9*ln(x))

3. Distribute the negative sign to prepare for differentiation.

x2+7*ln(3*x−2)−ln(7)−9*ln(x)

4. Differentiate each term with respect to x Use the chain rule for the term 7*ln(3*x−2) where d(ln(u))/d(x)=1/u⋅d(u)/d(x)

d(x2)/d(x)+(d(7)*ln(3*x−2))/d(x)−d(ln(7))/d(x)−(d(9)*ln(x))/d(x)

5. Calculate the derivatives of the individual components. Note that ln(7) is a constant, so its derivative is 0

2*x+7⋅1/(3*x−2)⋅3−0−9⋅1/x

6. Simplify the expression by multiplying the constants in the second term.

2*x+21/(3*x−2)−9/x

Final Answer

d()/d(x)*ln((e(x2)*(3*x−2)7)/(7*x9))=2*x+21/(3*x−2)−9/x

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