Introduction to the First Derivative

Find the first derivative formula for each of the following functions twice: first by evaluating $\lim\limits_{h\to0}\frac{\fe{f}{x+h}-\fe{f}{x}}{h}$ and then by evaluating $\lim\limits_{t\to x}\frac{\fe{f}{t}-\fe{f}{x}}{t-x}$.

$\fe{f}{x}=x^2$

Find the first derivative formula for each of the following functions twice: first by evaluating $\lim\limits_{h\to0}\frac{\fe{f}{x+h}-\fe{f}{x}}{h}$ and then by evaluating $\lim\limits_{t\to x}\frac{\fe{f}{t}-\fe{f}{x}}{t-x}$.

$\fe{f}{x}=\sqrt{x}$

It can be shown that $\lim\limits_{h\to0}\frac{\fe{\sin}{h}}{h}=1$ and $\lim\limits_{h\to0}\frac{\fe{\cos}{h}-1}{h}=0$. Use these limits to help you to establish the first derivative formula for $\fe{\sin}{x}$.

Suppose that an object is tossed into the air in such a way that the elevation of the object (measured in ft) $t$ seconds after the object was tossed is given by the function $\fe{s}{t}=150+60t-16t^2$.

Find the velocity function for this motion and use it to determine the velocity of the object 4.1 s into its motion.

Suppose that an object is tossed into the air in such a way that the elevation of the object (measured in ft) $t$ seconds after the object was tossed is given by the function $\fe{s}{t}=150+60t-16t^2$.

Find the acceleration function for this motion and use that function to determine the acceleration of the object 4.1 s into its motion.

Determine the unit on the first derivative function for each of the following functions. Remember, we do not simplify derivative units in any way, shape, or form.

$\fe{h}{t}$ is the elevation of the space shuttle (mi) where $t$ is the amount of time that has passed since lift-off (measured in s).

It can be shown that the derivative formula for the function $f$ defined by $\fe{f}{x}=\frac{2x^3+2x+1}{3+5x^2}$ is $\fe{\fd{f}}{x}=\frac{10x^4+8x^2-10x+6}{25x^4+30x^2+9}$.

Determine the equation of the tangent line to $f$ at $1$.