Find the Derivative - d/dx 3^(x natural log of x)

Problem

d()/d(x)*3(x*ln(x))

Solution

1. Identify the form of the function as au(x) where a=3 and u(x)=x*ln(x)

2. Apply the chain rule for exponential functions, which states d(au)/d(x)=au*ln(a)d(u)/d(x)

3. Differentiate the exponent u(x)=x*ln(x) using the product rule, (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

4. Calculate the derivative of the exponent: (d(x)*ln(x))/d(x)=x⋅1/x+ln(x)⋅1=1+ln(x)

5. Substitute the derivative of the exponent back into the chain rule formula.

6. Simplify the final expression.

Final Answer

d(3(x*ln(x)))/d(x)=3(x*ln(x))*ln(3)*(1+ln(x))

Want more problems? Check here!