Definition of Derivatives

The derivative of a function is its

• Slope
• Maximum/Minimum
• Instantaneous rate of change
• Common Denominator

f'(x)=

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

• $\lim_{x\rightarrow c}\left(\frac{f\left(x\right)-f\left(c\right)}{x-c}\right)$

• $\frac{\text{d}y}{\text{d}x}$

• an equation for the slope of the tangent line

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

•  $5x^2$  

•  $10x$  

• 10

• DNE

Tags

CCSS.HSA.APR.A.1

CCSS.HSA.SSE.A.2

CCSS.HSA.SSE.B.3

 $\lim_{x\rightarrow2}\left(\frac{x^2-4}{x-2}\right)$  

• 2

• 4

• 6

• 8

Tags

CCSS.HSA.APR.D.6

CCSS.HSA.APR.D.7

CCSS.HSA.SSE.A.2

CCSS.HSA.SSE.B.3

If a function f(x) has a derivative that is negative, what does that tell you?

• The function f(x) is increasing

• The function f(x) is decreasing

•  $f\left(x\right)<0$  

• Nothing other than  $f'\left(x\right)<0$  

What is the traditional definition of a derivative?

Evaluate responses using AI:

OFF

 $\lim_{x\rightarrow1}\ \frac{\sqrt{x^2+3x}-2}{x-1}$  is the instantaneous rate of change of what function at what value of the domain? 

•  $f\left(x\right)=\sqrt{x^2-3x}-2$  at  $x=1$  

•  $f\left(x\right)=\sqrt{x^2-3x}$  at  $x=2$  

•  $f\left(x\right)=\frac{\sqrt{x^2-3x}}{x-1}$  at  $x=1$  

•  $f\left(x\right)=\sqrt{x^2-3x}$  at  $x=1$  

Tags

CCSS.HSF.IF.A.2

The alternate definition of the derivative gives a derivative function while the traditional definition of the derivative gives a numerical derivative.

• True

• False

 $\lim_{h\rightarrow0}\ \frac{e^{\left(x+h\right)}-1}{h}$  This limit will compute the derivative of what f(x) function? At what x value?

Tags

CCSS.HSA.SSE.A.1

CCSS.HSF.LE.A.1

Which graph is the derivative of the top graph?

• A
• B
• C
• D

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7