Given that $\frac{dx}{dt}=4t^3$ and $x=5$ when $t=1$, what is the value of $x$ when $t=2$?

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 $x=3$?

What is the slope of the tangent line to the polar curve $r=4{\theta }^2$ at the point where $\theta =\frac{\pi }{4}$?

Let g be the function given by $g\left(x\right)=x^4-3x^3-x$. What are all values of $x$ such that $g^{\prime }\left(x\right)=12$?

Let $f\left(u\right)=u^2$ and $g\left(x\right)=3x^6+6$. Find $(f\circ g)\text{'}(-1)$. (Type an exact answer.)

Given the function $z=\sin (xy)$, which of the following is the correct expression for the partial derivative of $z$ with respect to $x$?