Determine the sign of the derivative at the indicated point.

Determine the sign of the derivative at the indicated point.

If $f\left(2\right)=3$ and  $f'\left(2\right)=-1$  , find the equation of the tangent line to  $f\left(x\right)$  at  $x=2$  . 

Find the equation of the tangent line to the graph of $f\left(x\right)=x^2-2x-3$ at $x=2$  . 

Differentiate $y=\frac{3x}{x^2+1}$  

Find the average rate of change for the function $f\left(t\right)=\frac{t^3+2}{t}$ on the interval [1,4] 

Find the instantaneous rate of change for the function $f\left(t\right)=\frac{t^3+2}{t}$ when  $t=2$  

Suppose that  $h\left(x\right)=\frac{g\left(x\right)}{f\left(x\right)}$  and  $g\left(2\right)=3,\ g'\left(2\right)=-1,\ f\left(2\right)=5,\ f'\left(2\right)=-2$ .  Find  $h'\left(2\right)$  

If $f\left(x\right)=2x^2+4$ , which of the following will calculate the derivative of  $f\left(x\right)$  ? 

Let $h\left(x\right)=f\left(x\right)\cdot g\left(x\right)$ .  Find   $h'\left(2\right)$  

What is $g'\left(3\right)$ ?

One lazy day, Uncle Si decides to sit under a shade tree next to a road by the swamp and count the number of ducks crossing this road at each hour after 9am. Determine the average number of ducks per hour which have crossed the road during Uncle Si's observation.

One lazy day, Uncle Si decides to sit under a shade tree next to a road by the swamp and count the number of ducks crossing this road at each hour after 9am. Estimate the value of  $f'\left(4.5\right)$  and explain its meaning.

 $\lim_{h\rightarrow0}\frac{\sin\left(x+h\right)-\sin x}{h}=$  

Differentiate $3f\left(x\right)-g\left(x\right)$ at  $x=-1$   

Differentiate $3f\left(x\right)\cdot g\left(x\right)$ at  $x=-1$   

Differentiate $\frac{f\left(x\right)}{g\left(x\right)+2}$ at $x=0$

The graph of f', the derivative of f, is shown for the interval (-2,7). What are all the values of x for which f has a horizontal tangent? Note this one is tricky and will not be on your test (for this unit).

Suppose that  $k\left(x\right)=2f\left(x\right)-3g\left(x\right)$  and  $g\left(2\right)=3,\ g'\left(2\right)=-1,\ f\left(2\right)=5,\ f'\left(2\right)=-2$ .  Find  $k'\left(2\right)$  

Let $m\left(x\right)=\frac{\tan x}{f\left(x\right)}$ .  Find   $m'\left(0\right)$  

Differentiate  $f\left(x\right)\cdot\cos x$  when  $x=0$   

Find g'(x) given $g\left(x\right)=\frac{2x+3}{3x^2-5}$ 

Find g'(x) given $g\left(x\right)=\frac{6\sin x}{\sqrt{x^3}}$  

Find f'(x) given $f\left(x\right)=\left(\sec x\right)\ln x$ 

Find f'(x) given

Find the average rate of change of $f\left(x\right)=x^2-3x+5$ on [-2,1]. 

 $\lim_{h\rightarrow0}\frac{\left[\left(x+h\right)^2-3\left(x+h\right)+5\right]-\left(x^2-3x+5\right)}{h}=$  

 $\lim_{h\rightarrow0}\frac{\csc\left(x+h\right)-\csc x}{h}$  

 $\lim_{x\rightarrow a}\frac{e^x-e^a}{x-a}=$  

 $\lim_{x\rightarrow a}\frac{\left[x^3+\ln x\right]-\left(a^3+\ln a\right)}{x-a}=$  

Find f'(x) for

Find g'(x) if $g\left(x\right)=\frac{5}{x^5}+\frac{x^5}{5}+\pi^2$  

Find f'(x) if $f\left(x\right)=x^{\frac{2}{3}}+e^{\frac{2}{3}}$  

Let $h\left(x\right)=x^2-g\left(x\right)$  .  Find  $h'\left(-1\right)$   

Let $k\left(x\right)=\sqrt{x}+3g\left(x\right)$  .  Find  $k'\left(4\right)$   

Let $m\left(x\right)=\frac{7}{3x}-g\left(x\right)+2x-6$  .  Find  $m'\left(7\right)$   

Given $h\left(x\right)=-2\cos x-f\left(x\right)$ .  Find  $h'\left(3\right)$   

Let $b\left(x\right)=e^x+2f\left(x\right)$ .  Find  $b'\left(0\right)$   

The function shown

The function shown

Find $$\frac{dy}{dx}$$ for  $$y=4e^x\cot x$$   

Find $$f'\left(\theta\right)$$ for  $$f\left(\theta\right)=\csc\theta\cot\theta$$   

If the function above is differentiable at x=3, find a+b

Find the value of b, if possible, that makes the function differentiable at x=2.

Find the value of a, if possible, that makes the function differentiable at x=1

If  $$y=x\sin x$$ , then  $$\frac{\text{d}y}{\text{d}x}$$  = 

Let f be the function given by $$f\left(x\right)=x^3-6x^2+8x-2$$ .  What is the instantaneous rate of change of f at x=3?

If $$k\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$$ , what is the value of  $$k'\left(3\right)$$  ? 

The graph of a twice-differentiable function f is shown in the graph. Which of the following is true?

The graph shows the function g and the line tangent to the graph at x=-1. Let $$h\left(x\right)=e^x\cdot g\left(x\right)$$ .  What is  $$h'\left(-1\right)$$  ?

What is the value of $$k+c$$   if f(x) is differentiable everywhere?

If $$\lim_{x\rightarrow3}f\left(x\right)=7$$ , which of the following must be true?

I. f is continuous at x=3

II. f is differentiable at x=3

III. f(3)=7 

$$\lim_{h\rightarrow0}\left(\frac{5\left(x+h\right)^2-5x^2}{h}\right)$$  

When you plug a given point into the derivative of a function, you get this.

This is a line that touches the function at a given point and has a slope that is equal to the instantaneous rate of change at a given point.

This is a line that touches the function at 2 given points.

$$\lim_{x\rightarrow a}\frac{\ln x-\ln a}{x-a}$$

$$\lim_{x\rightarrow a}\frac{\csc x-\csc a}{x-a}$$

$$\lim_{x\rightarrow\frac{\pi}{4}}\frac{\cos x-\cos\frac{\pi}{4}}{x-\frac{\pi}{4}}$$