Given the implicit equation $x^2+xy+y^3=1$, find the value of the third derivative $y^{\prime \prime \prime }$ with respect to $x$ at the point where $x=1$ and $y=0$.

In what direction(s) does the derivative of $f\left(x\right)$ provide information about the behavior of the function $f\left(x\right)$?

Given the equation $3+4e^{x+1}=11$, what is the value of $x$?

Which concept is fundamental in determining the derivative of a function as a new function?

Given $x=e^t$ and $y=t{e}^{-t}$, find $\frac{dy}{dx}$ and ${\frac{dy}{dx}}^2$ in terms of $t$.

Let $f\left(x\right)$ be the function given by $f\left(x\right)=x^3-2x+1$. What is the instantaneous rate of change of $f$ with respect to $x$ at $x=2$?

Given $\frac{\partial y}{\partial t}=6e^{-0.08(t-5{)}^2}$, by how much does $y$ change as $t$ changes from $t=1$ to $t=6$?

If $f\left(x\right)=\sin x+2x+1$ and $g$ is the inverse function of $f$, what is the value of $g^{\prime }\left(1\right)$?

If $f$ is a differentiable function and $y=\sin (f(x^2\left)\right)$, what is $\frac{dy}{dx}$ when $x=3$?