163k views
1 vote
Finding a Second Derivative, find the second derivative of the function.

Finding a Second Derivative, find the second derivative of the function.-example-1
User Fenway
by
8.8k points

1 Answer

3 votes

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}

User Sani Yusuf
by
8.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories