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.