To find the slope of the tangent line to the equation [tex]\(\cos(x - y) = x e^z\)[/tex] at the point [tex]\((0, \frac{5}{2})\)[/tex], we need to determine [tex]\(\frac{dy}{dx}\)[/tex] at this specific point. This process involves the following steps:
1. Implicit Differentiation: Differentiate the given equation [tex]\(\cos(x - y) = x e^z\)[/tex] with respect to [tex]\(x\)[/tex], treating [tex]\(y\)[/tex] as a function of [tex]\(x\)[/tex].
[tex]\[
\frac{d}{dx}[\cos(x - y)] = \frac{d}{dx}[x e^z]
\][/tex]
2. Apply Chain Rule: Differentiate each side with respect to [tex]\(x\)[/tex].
Let’s start with the left-hand side:
[tex]\[
\frac{d}{dx}[\cos(x - y)] = -\sin(x - y) \cdot \frac{d}{dx}[x - y]
\][/tex]
[tex]\[
= -\sin(x - y) \cdot \left(\frac{d}{dx}[x] - \frac{d}{dx}[y]\right)
\][/tex]
[tex]\[
= -\sin(x - y) \cdot (1 - \frac{dy}{dx})
\][/tex]
Now the right-hand side:
[tex]\[
\frac{d}{dx}[x e^z] = e^z \cdot \frac{d}{dx}[x] + x \cdot \frac{d}{dx}[e^z]
\][/tex]
[tex]\[
= e^z \cdot 1 + x \cdot 0 \quad (\text{because } e^z \text{ is a constant with respect to } x)
\][/tex]
[tex]\[
= e^z
\][/tex]
3. Combine Differentiated Results:
[tex]\[
-\sin(x - y) \cdot (1 - \frac{dy}{dx}) = e^z
\][/tex]
4. Solve for [tex]\(\frac{dy}{dx}\)[/tex]:
Distribute [tex]\(-\sin(x - y)\)[/tex]:
[tex]\[
-\sin(x - y) + \sin(x - y) \cdot \frac{dy}{dx} = e^z
\][/tex]
[tex]\[
\sin(x - y) \cdot \frac{dy}{dx} = e^z + \sin(x - y)
\][/tex]
[tex]\[
\frac{dy}{dx} = \frac{e^z + \sin(x - y)}{\sin(x - y)}
\][/tex]
[tex]\[
\frac{dy}{dx} = \frac{e^z}{\sin(x - y)} + 1
\][/tex]
Now simplify further to:
[tex]\[
\frac{dy}{dx} = \frac{e^z + \sin(x - y)}{\sin(x - y)}
\][/tex]
5. Evaluate at the Point [tex]\((0, \frac{5}{2})\)[/tex]:
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = \frac{5}{2}\)[/tex]:
[tex]\[
\frac{dy}{dx} \bigg|_{(0, \frac{5}{2})} = \frac{e^z - \sin(\frac{5}{2})}{\sin(\frac{5}{2})}
\][/tex]
Thus, the slope of the tangent line at the given point is:
[tex]\[
-(e^z - \sin(\frac{5}{2}))/\sin(\frac{5}{2})
\][/tex]
From our provided choices, this does not match any of the listed options (0, 1, 2, -2). Therefore, the correct answer is:
None of these.
