Obtain the derivative of 𝑧=(2𝑥-𝑦)(𝑥+3𝑦) with respect to yQuestion 10Answera.5𝑥-6𝑦b.(2𝑥-1)(𝑥+3)c.4𝑥+5𝑦d.(2-𝑦)(1+3𝑦)

Step 1: Break Down the Problem

We need to find the derivative of the function $z = (2x - y)(x + 3y)$ with respect to $y$. This will involve using the product rule for differentiation, which states that if you have a function $u(y)v(y)$, the derivative is given by:

$\frac{d}{dy}(uv) = u'v + uv'$

Step 2: Relevant Concepts

1. Let $u = 2x - y$ and $v = x + 3y$.
2. Then, we need to find the derivatives $u'$ and $v'$:
   • $u' = \frac{d}{dy}(2x - y) = -1$
   • $v' = \frac{d}{dy}(x + 3y) = 3$

Step 3: Analysis and Detail

Now applying the product rule:

$\frac{dz}{dy} = u'v + uv'$ Substituting in the derivatives and the functions:

$\frac{dz}{dy} = (-1)(x + 3y) + (2x - y)(3)$

Calculating each term separately:

1. The first term is: $-1 \cdot (x + 3y) = -x - 3y$

2. The second term is: $(2x - y)(3) = 6x - 3y$

Combining both terms:

$\frac{dz}{dy} = (-x - 3y) + (6x - 3y)$ This simplifies to:

$\frac{dz}{dy} = 5x - 6y$

Step 4: Verify and Summarize

The derivative $\frac{dz}{dy}$ is calculated correctly as $5x - 6y$.

Final Answer

The final result is:

$\frac{dz}{dy} = 5x - 6y$

So the correct option is: a. 5x - 6y