Find the Derivative - d/d@VAR

Problem

d()/d(x)*ln(3*e(2*x−5)*(3*x3+5)7)

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) and the power rule ln(an)=n*ln(a)

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 inverse property ln(eu)=u and moving the exponent in front of the third term.

ƒ(x)=ln(3)+(2*x−5)+7*ln(3*x3+5)

3. Differentiate each term with respect to x Note that ln(3) and −5 are constants, so their derivatives are 0

d(ƒ)/d(x)=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)

d(ƒ)/d(x)=0+2+7⋅1/(3*x3+5)⋅d(3*x3+5)/d(x)

5. Calculate the derivative of the inner function 3*x3+5 which is 9*x2

d(ƒ)/d(x)=2+7/(3*x3+5)⋅9*x2

6. Multiply the constants in the second term to find the final derivative.

d(ƒ)/d(x)=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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