Question

If f(5)=3 and f’(5)=-2, find the derivative of x^2 f(x) at x=5

Answer 1

For this question we will use the following rule for derivatives:

$(g\mleft(x\mright)f(x))^{\prime}=g^{\prime}(x)f(x)+g(x)f^{\prime}(x)\text{.}$

Therefore:

$\begin{gathered} (x^2f(x))^{\prime}=(x^2)^{\prime}f(x)+x^2f^{\prime}(x), \\ (x^2f(x))^{\prime}=2xf(x)+x^2f^{\prime}(x)\text{.} \end{gathered}$

Since f(5)=3, and f'(5)=-2, we get that the derivative of x²f(x) at x=5 is:

$2\cdot5f(5)+5^2f^{\prime}(5)=10\cdot3+25(-2)=30-50=-20.$

Answer: Option d.