Find the Derivative - d/d@VAR f(x)=e^(3x^3+1) natural log of 2x^3+3

Problem

d()/d(x)*(e(3*x3+1)*ln(2*x3+3))

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, u(x)=e(3*x3+1) and v(x)=ln(2*x3+3) apply the product rule: d()/d(x)*(u*v)=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first function u(x)=e(3*x3+1) using the chain rule.

d(e(3*x3+1))/d(x)=e(3*x3+1)⋅d(3*x3+1)/d(x)

d(e(3*x3+1))/d(x)=9*x2*e(3*x3+1)

3. Differentiate the second function v(x)=ln(2*x3+3) using the chain rule.

d(ln(2*x3+3))/d(x)=1/(2*x3+3)⋅d(2*x3+3)/d(x)

d(ln(2*x3+3))/d(x)=(6*x2)/(2*x3+3)

4. Combine the parts using the product rule formula.

d(ƒ(x))/d(x)=e(3*x3+1)⋅(6*x2)/(2*x3+3)+ln(2*x3+3)⋅9*x2*e(3*x3+1)

5. Factor out the common terms 3*x2*e(3*x3+1) to simplify the expression.

d(ƒ(x))/d(x)=3*x2*e(3*x3+1)*(2/(2*x3+3)+3*ln(2*x3+3))

Final Answer

d()/d(x)*(e(3*x3+1)*ln(2*x3+3))=3*x2*e(3*x3+1)*(2/(2*x3+3)+3*ln(2*x3+3))

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