Use the definition of the derivative to $f\,'(x)$, when $\displaystyle{f\,(x) = \frac{1}{2-3x}}$

Find $f\,'(x)$ using the definition of the derivative, where $\displaystyle{f\,(x)=\frac{1}{\sqrt{4-x}}}$, and then find the equation of the normal line at $x=3$

Find $f\,'(x)$ using the definition of the derivative, where $f\,(x)=x^2-4x+4$ and then determine where the tangent line to the graph of $y=f\,(x)$ is horizontal

Given $f\,'(x) = 3x^2-2x+1$

- Use the definition of the derivative to find $f\,'(x)$
- Find the equation of the tangent line to the graph of $y=f\,'(x)$ at $x=2$
- Find the point on the graph of $y=f\,'(x)$ where the tangent line is horizontal
- Find the equation of the normal line to the graph of $y=f\,'(x)$ at $x=-1$

A ball is thrown upward from the top of a building. The initial height is 640 ft, and the initial velocity (upward) is 64 ft/s. Its height above the ground is given by $h(t)=-16t^2+64t+640$. Use the definition of the derivative to answer the following questions:

- What is the instantaneous velocity at $t=1$ second?
- What is the height of the ball at $t=1$ second?
- When will the ball reach its maximum height?

The cost (in dollars) to make $x$ new computers is given by $C(x) = x^2+196x+400$. The expected revenue is given by $R(x)=-3x^2+660x$. Use the definition of the derivative to answer the following questions:

- How many units need to be made in order to maximize profits?
- What is the maximum profit that can be made?

Use the definition of the derivative to find the derivative of each function given

$\displaystyle{f\,(x) = \frac{1}{\sqrt{1-2x}}}$

$\displaystyle{h(x)=6-\sqrt{x+4}}$

$\displaystyle{f(x)=\frac{1}{3}}$

$\displaystyle{f(x)=\frac{3}{2x+1}}$

$\displaystyle{g(x)=\frac{x}{x+1}}$

$\displaystyle{h(x)=1+\sqrt{x}}$

For each function given below, find $f\,'(x)$ using the definition of the derivative, then find the equation of the tangent line to the graph of $y=f\,(x)$ at the given $x$-value.

$\displaystyle{f\,(x) = \frac{1}{\sqrt{3x}}; \quad x=3}$

$\displaystyle{f\,(x) = \frac{1}{1-2x}; \quad x=1}$

$\displaystyle{f\,(x) = \frac{1}{x^2+1}; \quad x=-1}$

Given $\displaystyle{f\,(x) = \frac{2}{4-x}}$

- Find $f\,'(x)$ using the definition of the derivative
- Find the equation of the normal line to the graph of $y=f\,(x)$ at $x=2$.

Use the definition of the derivative to show that $f\,'(x) = \displaystyle{\frac{-1}{2x\sqrt{x}}}$ when $\displaystyle{f\,(x) = \frac{1}{\sqrt{x}}}$, and then find the equation of the normal line to the graph of the equation $y=f\,(x)$ at $x=4$

Given $\displaystyle{f\,(x) = \frac{1}{1+x}}$

- Use the definition of the derivative to find $f\,'(x)$
- Find the equation of the tangent line to the graph of $y=f(x)$ at $x=0$
- Find the equation of the normal line to the graph of $y=f(x)$ at $x=-2$