2. Intro to Derivatives

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

Suppose the function $f\left(x\right)$ has the derivative $f^{\prime }\left(x\right)=3x^2-4x+2$. Find the values of $f^{\prime }\left(1\right)$ and $f^{\prime }\left(2\right)$.

In the context of derivatives as functions, what does the derivative of a function represent?

Consider a function $y=f\left(x\right)$. Which of the following best defines the slope of this function at a point $x=a$?

Find the derivative of the function: $y=arctan\frac{1-x}{1+x}$.

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 $f\left(x\right)=x^2$, which of the following is the average rate of change of $f\left(x\right)$ on the interval from $x=3$ to $x=6$?

Given the function $g\left(x\right)={\sinh \left(x\right)}^2$, find its derivative $g\text{'}\left(x\right)$.

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

For the function $f\left(x\right)$, in which direction(s) does the derivative $f^{\prime }\left(x\right)$ provide information about the behavior of $f\left(x\right)$?

If $f\left(x\right)={(2x^2+5)}^7$, then which of the following is $f^{\prime }\left(x\right)$?

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 $f\left(x\right)$?

What is the derivative of the function $f\left(x\right)=e^{4x}$ with respect to $x$?

Suppose the figure above shows the graph of $f^{\prime }$, the derivative of a function $f$. At which of the following $x$-values does $f$ have a local maximum?

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