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Worked solution

Differentiate a polynomial-logarithm product

Differentiate x3ln⁡(2x+1)x^3\ln(2x+1) with respect to xx.

CJC · Discussion · Question 17

Answer

2x32x+1+3x2ln⁡(2x+1)\dfrac{2x^3}{2x+1}+3x^2\ln(2x+1).

Apply the product and chain rules

With y=x3ln⁡(2x+1)y=x^3\ln(2x+1), dydx=x3(22x+1)+ln⁡(2x+1)(3x2)=2x32x+1+3x2ln⁡(2x+1)\dfrac{dy}{dx}=x^3\left(\dfrac2{2x+1}\right)+\ln(2x+1)(3x^2)=\dfrac{2x^3}{2x+1}+3x^2\ln(2x+1), for x>−1/2x>-1/2.

How did this part go?

Common mistakes

The derivative of ln(2x + 1) includes the inner derivative 2.

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