Given the graph of a function $f(x,y)$, at the point $(a,b)$, the surface is increasing as $x$ increases and decreasing as $y$ increases. Which of the following correctly describes the signs of the partial derivatives $f_x(a,b)$ and $f_y(a,b)$?

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 $x$?

Find the first partial derivatives of the function $z=x\sin (xy)$ with respect to $x$ and $y$.

Given that f and g are differentiable functions, which of the following correctly expresses the derivative of $v\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ in terms of $f$, $g$, $f^{\prime }$, and $g^{\prime }$?

Given $y=x\ln \left(x\right)$, which of the following correctly gives both the first and second derivatives, $y^{\prime }$ and $y^{″}$?

Given the function $f\left(x\right)=\sin \left(x\right)\ln \left(x^4\right)$, compute $f^{\prime }\left(x\right)$. Which of the following is correct?

Given the function $f(x,y)=ax+by+cx+dy$, where $a$, $b$, $c$, and $d$ are constants, find the first partial derivatives $f_x$ and $f_y$.