All About Derivatives!

 What is the derivative of the function:

 $f\left(x\right)=x^{100}$ 

•  $x^{100}-x^{99}$  

•  $\frac{x^{101}}{101}$  

•  $100x^{99}$  

•  $99x^{100}$  

What can a derivative NOT do?

• Find the area underneath a graph on any closed or open interval

• Calculate the instantaneous rate of change at a point

• Discover the minimum and maximum values of a function

• Create tangent and normal lines at a point

What test can we utilize to find an inflection point(s) of a function?

Find the following minimum and maximum values of the function on the provided interval (click all options that are the answers!):

 $g\left(x\right)=-\left(2x+5\right)^3$ 

Interval:  $\left(-\infty,\ \infty\right)$  

• No Absolute Minimum

•  $\left(-\frac{5}{2},\ 0\right)$  

•  $\left(-3,\ 1\right)$  

• No Absolute Maximum

•  $\left(0,\ -125\right)$  

Find the second derivative of the following function:  $h\left(x\right)=\cos\left(\cot\left(x\right)\right)$  

•  $-\csc^2\left(x\right)\left(2\cot x\sin\left(\cot\left(x\right)\right)+\csc^2\left(x\right)\cos\left(\cot\left(x\right)\right)\right)$  

•  $-2\csc^3\left(x\right)\cot\left(x\right)\sin\left(\cot\left(x\right)\right)+\csc^5\left(x\right)\cos\left(\cot\left(x\right)\right)$  

•  $\csc^2x\cdot\sin\left(\cot\left(x\right)\right)$  

•  $-\csc^3\left(x\right)\left(2\cot x\sin\left(\cot\left(x\right)\right)\right)$  

Using the provided information in the image above (click to enlarge), calculate the equations of the normal and tangent lines at the point provided. (NOTE: If you wish you may find that value in fractional form if you prefer not to work with decimals, however the provided answers will be in decimals, also be sure to round all answers to the nearest fifth decimal place).

• $Tangent:\ y=-0.10143x+0.84694$ $Normal:\ y=2325.5814x-1643.33945$

• $Tangent:\ y=-0.00043x+0.84690$ $Normal:\ y=2325.5814x-1643.33945$

• $Tangent:\ y=2325.5915x-1600.3295$ $Normal:\ y=-1020.03043x+849.0234$