2. Intro to Derivatives

Find the derivative of the function $h\left(x\right)=x^2\sin \left(x\right)$. Which of the following is correct?

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 $f_y(a,b)$?

Which function $f\left(x\right)$ and value $a$ correspond to the following limit representing the derivative of $f$ at $a$?$$\underset{h\to 0}{lim}\frac{{(3+h)}^2-9}{h}$$

The slope of the tangent line at a point on a function calculates which of the following?

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

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

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

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 $x=2$?

Which of the following best describes the derivative of a function $f\left(x\right)$?

Find the derivative of the function: $y={cot(\sin (x\left)\right)}^2$

Find the derivative of the function: $y=e^{tan\left(x\right)}$.

Given the function $f\left(x\right)=e^x\cosh \left(x\right)$, which of the following is the correct expression for $f^{\prime }\left(x\right)$?

Find the derivative of the function: $y=arctan\frac{1+x}{1-x}$. Which of the following is correct?

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

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 $f\left(x\right)$?