Derivatives Review 1

Basic Derivatives

100

Find the derivative of y= -3x⁷ 

y' = -21x⁶

200

Find the derivative of y=sin(x)-cos(x)

y' = cos(x) + sin(x)

300

Find the derivative of -x^1/3

y' = -(1/3)x^-2/3

400

Find the derivative of y = 1/x + 2/3 + 1/x^2/3

y' = -1/x² - 2/3x^5/3

500

Find the derivative of y= x^-1/2 + 23 - e^x

y' = (-1/2)x^-3/2 - e^x

Product Rule 1

100

Find the derivative of y = x⁴(x³ + 2x² +5)

y' = 7x⁶ + 12x⁵ +20x³

200

Find the derivative of y = x³e^x

y' = x³e^x + 3x²e^x

300

Find the derivative of y = sin(x)ln(x)

y' = sin(x)/x + cos(x)ln(x)

400

Find the derivative of y=tan(x)sec(x)

y' = tan²(x)sec(x) + sec³(x)

500

Find the derivative of y=sin(x)cos(x)

y' = -sin²(x) + cos²(x)

Misc.

100

Find the derivative of f(x) = (x-2)(x-1) at x=-1

f'(x)= 2x - 3

f'(-1) = -5 

200

Find the 44th derivative of y=100-x⁴³

y' = 0

300

Find the slope of the secant line for the equation y= -x² + 3x between x=-2 and x=1.

slope = 4

400

Find the equation of the secant line for the equation y= -x² + 3x between x=-2 and x=1.

y = 4x - 2

500

Find the 99^th derivative of y=x⁹⁹

y' = 99!

Higher Derivatives

100

Find the second derivative of y = 3x³ - x² +12

y'' = 18x - 2

200

Find the third derivative of y = -sin(x)

y''' = cos(x)

300

Find the second derivative of y=ln(x)

y'' = -x^-2

400

Find the 14 derivative of y = sin(x)

y⁽¹⁴⁾ = -sin(x)

500

Find the 99^th derivative of y=e^x

y⁽⁹⁹⁾ = e^x

Graphs of the Derivative

100

If f'(x) < 0 for an interval, what does that tell you about the graph of f(x) at that interval?

f(x) is decreasing

200

How is the derivative graph-related to the original graph?

Derivative graph tells us the slope and where we are increasing/decreasing

300

If f(x)=x³, sketch the derivative. What is true about the graph of f(x)?

Graph of 3x² - f(x) >= 0

400

Based on the graph of y = -x⁷, what do we know about y'?

y' <= 0 for all real numbers

500

Sketch y,y',y'' for the function y=x²

y=x²

y=2x

y=2