Question

Finding a Second Derivative, find the second derivative of the function.

Answer 1

Given the function f(x):

`f(x)=x\cos x`

The product rule for the derivative states that:

`\begin{gathered} f(x)=g(x)* h(x) \\ \\ \Rightarrow f^(\prime)(x)=g^(\prime)(x)* h(x)+g(x)* h^(\prime)(x) \end{gathered}`

Using this rule, we calculate the first derivative of f(x):

`\begin{gathered} f^(\prime)(x)=(x)^(\prime)\cos x+x(\cos x)^(\prime)=(1)\cos x+x(-\sin x) \\ \\ \Rightarrow f^(\prime)(x)=\cos x-x\sin x \end{gathered}`

We take the derivative one more time to calculate the second derivative:

`\begin{gathered} f^(\prime)^(\prime)(x)=(\cos x)^(\prime)-(x)^(\prime)\sin x-x(\sin x)^(\prime)=(-\sin x)-(1)\sin x-x(\cos x) \\ \\ \therefore f^(\prime)^(\prime)(x)=-2\sin x-x\cos x \end{gathered}`