The question and problem in the image relate to finding the derivative of one function in terms of another when two parametric equations are given:

1. $x(t) = \sin(t) - \cos(t)$
2. $y(t) = \sin(t) + \cos(t)$

We need to express $\frac{dy}{dx}$ in terms of these parametric functions.

Steps:

1. Find $\frac{dx}{dt}$: $x(t) = \sin(t) - \cos(t)$ Differentiating with respect to $t$: $\frac{dx}{dt} = \cos(t) + \sin(t)$

2. Find $\frac{dy}{dt}$: $y(t) = \sin(t) + \cos(t)$ Differentiating with respect to $t$: $\frac{dy}{dt} = \cos(t) - \sin(t)$

3. Express $\frac{dy}{dx}$: By the chain rule: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ Substituting: $\frac{dy}{dx} = \frac{\cos(t) - \sin(t)}{\cos(t) + \sin(t)}$

Simplification:

The derivative $\frac{dy}{dx}$ is expressed in terms of $t$, the parameter. If needed, this can be converted back into $x$ and $y$ by eliminating $t$, but it depends on further context.

Would you like me to simplify further, or provide additional details about using this result?

Related Questions:

1. How can we eliminate $t$ to find a direct relationship between $x$ and $y$?
2. How does the parametric derivative relate to the tangent line slope?
3. Can this result be used for polar coordinate conversion?
4. What are the critical points of $\frac{dy}{dx}$?
5. How would the second derivative $\frac{d^2y}{dx^2}$ be calculated?

Tip: Always differentiate carefully with respect to the parameter $t$, and ensure correct substitution when transitioning between parameterized and explicit forms of equations.