2. Intro to Derivatives

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 $x$?

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 $f^{\prime }\left(2\right)$.

The Reciprocal Rule

b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule.

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 $g\text{'}\left(x\right)$.

Using the Alternative Formula for Derivatives

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.

g(x) = 1 + √x