Given the equation $e^x+y=1$, explain how this equation implicitly defines a function and discuss the relationship between $x$ and $y$ without solving for $y$ explicitly.

Consider the equation $\sin (x)+y^2=4$. Explain how this equation implicitly defines a function and describe the relationship between $x$ and $y$ without solving for $y$ explicitly.

Given the equation $x^3+y^3=6xy$, explain how this equation implicitly defines a function and describe the relationship between $x$ and $y$ without solving for $y$ explicitly.

Consider the equation $\ln (x)+y=3$. Explain how this equation implicitly defines a function and describe the relationship between $x$ and $y$ without solving for $y$ explicitly.

Given the implicitly defined function $x^2+y^2=25$, use implicit differentiation to find $\frac{dy}{dx}$.

For the implicitly defined function $x^3+y^3=6xy$, use implicit differentiation to find $\frac{dy}{dx}$.

Given the implicitly defined function $e^x+y^2=xy$, use implicit differentiation to find $\frac{dy}{dx}$.

For the implicitly defined function $\sin (x)+\cos (y)=1$, use implicit differentiation to find $\frac{dy}{dx}$.

Given the implicitly defined function $x^{2}y+y^3=4$, use implicit differentiation to find $\frac{dy}{dx}$.

Consider the implicit function defined by the equation $x^2+y^2=25$. Describe the relationship between $x$ and $y$ and explain why this relationship is not easily expressed in explicit form.

Given the implicit function $x^2-xy+y^2=7$, find the points where the curve intersects the x-axis.

For the implicit function defined by $x^3+y^3=6xy$, determine if the point $(1,2)$ lies on the curve.

Given the implicit function $\sin (x)+\cos (y)=1$, find the value of $y$ when $x=\frac{\pi }{2}$.

For the implicit function $e^x+y^2=4$, determine the slope of the tangent line at the point $(0,2)$.