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Calculating Derivative – A multiplication of polynom, ln and e – Exercise 6275

Exercise

Find the derivative of the following function:

f(x)=x^2\cdot e^{3x}\cdot \ln(2x)

Final Answer


f'(x)=xe^{3x}(2\ln(2x)+3x\ln(2x)+1)

Solution

We simplify the function before differentiating:

f(x)=x^2\cdot e^{3x}\cdot \ln(2x)=

=(x^2\cdot e^{3x})\cdot \ln(2x)

Using Derivative formulas and the multiplication rule in Derivative Rules, we get the derivative:

f'(x)=(2xe^{3x}+x^2\cdot 3e^{3x})\ln (2x)+(x^2e^{3x}\cdot\frac{1}{2x}\cdot 2=

One can simplify the derivative:

=2xe^{3x}\ln(2x)+3x^2e^{3x}\ln(2x)+xe^{3x}=

=xe^{3x}(2\ln(2x)+3x\ln(2x)+1)

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