Find the Derivative - d/dx natural log of 3e^(2x-5)(3x^3+5)^7

Problem

d(ln(3*e(2*x−5)*(3*x3+5)7))/d(x)

Solution

1. Apply logarithm properties to simplify the expression before differentiating. Use the product rule for logarithms ln(a*b*c)=ln(a)+ln(b)+ln(c)

ln(3*e(2*x−5)*(3*x3+5)7)=ln(3)+ln(e(2*x−5))+ln((3*x3+5)7)

2. Simplify further using the power rule for logarithms ln(xn)=n*ln(x) and the inverse property ln(eu)=u

ln(3)+(2*x−5)+7*ln(3*x3+5)

3. Differentiate the sum term by term. The derivative of a constant is 0

d(ln(3))/d(x)+d(2*x−5)/d(x)+(d(7)*ln(3*x3+5))/d(x)

4. Apply the chain rule to the logarithmic term. The derivative of ln(u) is 1/u⋅d(u)/d(x)

0+2+7⋅1/(3*x3+5)⋅d(3*x3+5)/d(x)

5. Calculate the derivative of the inner polynomial 3*x3+5

d(3*x3+5)/d(x)=9*x2

6. Combine the results into the final expression.

2+(7⋅9*x2)/(3*x3+5)

7. Simplify the fraction by multiplying the constants in the numerator.

2+(63*x2)/(3*x3+5)

Final Answer

d(ln(3*e(2*x−5)*(3*x3+5)7))/d(x)=2+(63*x2)/(3*x3+5)

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