Question

Explain the steps because I don’t understand the work. (Work and answer are shown) #18g'(0)=-1 and g(0)= pi/2

Answer 1

Step-by-step explanation

Step 1

`\begin{gathered} \text{the product rule is } \\ y=\text{ f(x)}\cdot g(x) \\ y^(\prime)=f^(\prime)(x)g(x)+f(x)g^(\prime)(x) \\ \end{gathered}`

we have.

step 2

`w(x)=g(x)\sin (x)`

then

`\begin{gathered} w^(\prime)(x)=g^(\prime)(x)\sin (x)+g(x)\cos (x) \\ \end{gathered}`

Also,

`g^(\prime)(0)=-1\text{ and g(0)}`

replace

when x=0 it is w(0)

`\begin{gathered} w^(\prime)(x)=g^(\prime)(x)\sin (x)+g(x)\cos (x) \\ w^(\prime)(0)=g^(\prime)(0)\sin (0)+g(0)\cos (0) \\ w^(\prime)(0)=-1\cdot(0)+(\pi)/(2)\cdot1 \\ w^(\prime)(0)=0+(\pi)/(2) \\ w^(\prime)(0)=(\pi)/(2) \end{gathered}`