Question

(a) z=f(x)g(y)

In this equation, z is a function of both x and y, and it's defined as the product of two functions: f(x) and g(y).
∂z/∂x

Answer 1

To find the partial derivative of z with respect to x in the equation z=f(x)g(y), use the product rule of differentiation.

Step-by-step explanation:

In the equation $z = f(x)g(y)$, where $z$ is a function of both $x$ and $y$, and it's defined as the product of two functions $f(x)$ and $g(y)$, we can find $\frac{{\partial z}}{{\partial x}}$.

To find $\frac{{\partial z}}{{\partial x}}$, we need to differentiate $z$ with respect to $x$ while treating $y$ as a constant.

Since $z = f(x)g(y)$, we can apply the product rule of differentiation.

1. Differentiate $f(x)$ with respect to $x$ while treating $y$ as a constant. Let's call this derivative $f'(x)$.
2. Differentiate $g(y)$ with respect to $x$ while treating $y$ as a constant. This derivative is $0$ since $g(y)$ doesn't depend on $x$.
3. Multiply $f'(x)$ and $g(y)$.

So, $\frac{{\partial z}}{{\partial x}} = f'(x)g(y)$.