If 𝑧=𝑠𝑖𝑛⁡(3𝑥+2𝑦), find its derivative with respect to xQuestion 7Answera.𝑐𝑜𝑠⁡(3+2𝑦)b.3𝑐𝑜𝑠⁡(3𝑥+2𝑦)c.𝑠𝑖𝑛⁡(3+2𝑦)d.3𝑠𝑖𝑛⁡(3+2𝑦)

1. Break Down the Problem

We need to find the derivative of the function $z = \sin(3x + 2y)$ with respect to $x$. This involves applying the chain rule.

2. Relevant Concepts

The chain rule states that if you have a composite function $f(g(x))$, the derivative is given by: $\frac{df}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$ For $z = \sin(u)$ where $u = 3x + 2y$, we will use:

• $\frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx}$

3. Analysis and Detail

1. Identify $u$: $u = 3x + 2y$

2. Compute $\frac{du}{dx}$: $\frac{du}{dx} = \frac{d}{dx}(3x + 2y) = 3$ (Note that $y$ is treated as a constant with respect to $x$)

3. Apply the chain rule: $\frac{dz}{dx} = \cos(u) \cdot \frac{du}{dx} = \cos(3x + 2y) \cdot 3$

4. Verify and Summarize

The derivative of $z$ with respect to $x$ simplifies to: $\frac{dz}{dx} = 3\cos(3x + 2y)$ This corresponds to option b from the provided choices.

Final Answer

$\frac{dz}{dx} = 3\cos(3x + 2y) \quad \text{(Option b)}$