Find equations of the tangent line and normal line to the curve $y=14 \cos x$ at the point $(\pi / 3,7)$.
The derivative $y^{\prime}(x)=$
The slope of the tangent line is $m_{1}=$
The equation of the tangent line is $y=$
The slope of the normal line is $m_{2}=$
The equation of the normal line is $y=$

Step 1 ：Given the equation for y as a function of x, we have \(y = 14\cos(x)\).

Step 2 ：The derivative of y with respect to x is \(dy/dx = -14\sin(x)\).

Step 3 ：The slope of the tangent line to the curve at a given point is equal to the derivative of y with respect to x, which is \(dy/dx\).

Step 4 ：Evaluating this derivative at \(x = \pi/3\), we find that \(dy/dx = -14\sin(\pi/3) = -14\sqrt{3}/2 = -7\sqrt{3}\), so the slope of the tangent line is \(m_1 = dy/dx = -7\sqrt{3}\).

Step 5 ：The coordinates of the point on the curve are \(x = \pi/3\) and \(y = 7\).

Step 6 ：The equation of a line is given by \(y = mx + c\), where m is the slope and c is the y-intercept. Substituting the slope and the coordinates of the point into this equation, we can solve for c to find that \(c = y - mx = 7 - (-7\sqrt{3})(\pi/3) = 7 + 7\pi\sqrt{3}/3\).

Step 7 ：So, the equation of the tangent line is \(y = -7\sqrt{3}x + 7 + 7\pi\sqrt{3}/3\).

Step 8 ：The slope of the normal line is the negative reciprocal of the slope of the tangent line, so \(m_2 = -1/m_1 = 1/(7\sqrt{3}) = \sqrt{3}/21\).

Step 9 ：Substituting the slope of the normal line and the coordinates of the point into the equation for a line, we can solve for the y-intercept to find that \(c = y - m_2x = 7 - (\sqrt{3}/21)(\pi/3) = 7 - \pi\sqrt{3}/63\).

Step 10 ：So, the equation of the normal line is \(y = \sqrt{3}/21x + 7 - \pi\sqrt{3}/63\).

Step 11 ：\(\boxed{\text{Final Answer: The derivative } y'(x) = -14\sin(x), \text{ the slope of the tangent line is } m_1 = -7\sqrt{3}, \text{ the equation of the tangent line is } y = -7\sqrt{3}x + 7 + 7\pi\sqrt{3}/3, \text{ the slope of the normal line is } m_2 = \sqrt{3}/21, \text{ and the equation of the normal line is } y = \sqrt{3}/21x + 7 - \pi\sqrt{3}/63}\)