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

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)

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