2. Intro to Derivatives

Consider the function $f\left(x\right)$ whose second derivative is $f^{″}\left(x\right)=6x$. If $f\left(0\right)=2$ and $f^{\prime }\left(0\right)=3$, what is $f\left(x\right)$?

What is the derivative of the function $f\left(x\right)=e^{3x}$ with respect to $x$?

Given $f(x,y)={\sin }^2(mx+ny)$, which of the following is the correct expression for the second mixed partial derivative $\frac{{\partial }^{2}f}{\partial x\partial y}$?

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

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

Given that $3x-tany=4$, what is $\frac{dy}{dx}$ in terms of $y$?

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 $f\left(x\right)$?

Which of the following best describes the derivative of a function $f$ at a point $x=a$?

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

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 $t=6$?

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 $g^{\prime }$?

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

Find the gradient vector field of the function $f(x,y)=x{e}^{9xy}$. Which of the following is the correct gradient vector field $\nabla f(x,y)$?

What is the derivative of the function $f\left(x\right)=2e^x$ with respect to $x$?