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 that the second derivative of a function is $f^{″}\left(x\right)=2x-\cos \left(x\right)$, which of the following is a possible form for the original function $f\left(x\right)$?

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

Let $f$ be a differentiable function with derivative $f^{\prime }$. Which of the following statements is true?

Which of the following statements is not related to $differentiation$?

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 $t$.

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 $y=0$.