14) If \( y=\sin x+\sec x, x=3 \pi z \), then \( \frac{d y}{d z}=\ldots \ldots \ldots \). At \( z=1 \) \( \begin{array}{llll}\text { a) } 3 \pi & \text { b) }-3 \pi & \text { c) }-6 \pi & \text { d) }-1\end{array} \) b

To find \(\frac{dy}{dz}\), we first need to use the chain rule, knowing that \( x = 3 \pi z \). We can express \( \frac{dy}{dz} \) as: \[ \frac{dy}{dz} = \frac{dy}{dx} \cdot \frac{dx}{dz} \] Calculating \(\frac{dx}{dz}\), we have \( \frac{dx}{dz} = 3\pi \). Now we need to differentiate \( y = \sin x + \sec x \): \[ \frac{dy}{dx} = \cos x + \sec x \tan x \] Putting it all together at \( z = 1 \) (thus \( x = 3\pi \)), we substitute \( x = 3\pi \): \[ \frac{dy}{dx}\bigg|_{x=3\pi} = \cos(3\pi) + \sec(3\pi) \tan(3\pi) = -1 + 1 \cdot 0 = -1 \] Thus, \[ \frac{dy}{dz} = (-1) \cdot (3\pi) = -3\pi \] So, the final answer is \(-3\pi\). Now let’s add some lively context: Just imagine a trigonometric soirée! Sine and secant are dancing a wild tango while \( z \), the secret master of ceremonies, shifts the music tempo around. The relationship they share oscillates beautifully, and every twist tells the tale of calculus and derivatives! Next time you see a sine wave, think of the hidden connections dancing just beneath its surface. And if you're feeling curious, delve into the world of calculus further—a whole universe of functions and their derivatives awaits! You could explore the wonderful realm of differential equations or even the fascinating applications of calculus in engineering and physics, where even the simplest functions can lead to significant breakthroughs!