derivative rules (copy)

y = c

y' = 0

y = ax

y' = a

y = x^n

y' = nx^n-1

y = sinx

y' = cosx

y = cosx

y' = -sinx

y = tanx

y' = sec^2x

y = secx

y' = secxtanx

y = cscx

y' = -cscxcotx

y = cotx

y' = -csc^2x

y= a/b

y= b(a’) - a(b’) / (b)(b)

y= a*b

y=a(b’) + b(a’)

How to find out if a given function has any horizontal tangent lines

horizontal tangent line means the slope=0

1) take y’

2) set it equal to 0 and solve (calc)

how to find the points of a curve of a given function where the tangent line is parallel to the X AXIS

parallel to the x axis means slope=0

find y’

solve for x to find x points

then plug in to original function to find y points

how to find tangents to the curve at all points when the slope is a number #

the slope = a number (x does not = this number)

find y’

set it equal to the given number to solve for x

then plug in to the original function to find the points

how to find the x and y intercepts of the line that is tangent to the curve (insert function) at the point (a,b)

1) find y’

2) take the x from the given point and plug in to the y’ equation

3) write the tangent line equation using the slope found after plugging in x to y’

4) plug in 0 for y to find the x intercept

5) write the tangent line equation using the slope found after plugging in x to y’

4) plug in 0 for x to find the y intercept

to be differentiable at a point the ……

left and right derivatives must be equal at the point

how to solve for a and be in order for g(x) to be both continuous and differentiable at x=#

1) set the two equations in the piecewise function equal to each other

2)take the derivatives of each equation and solve for the left over variable. (a or b)

now u have a (or b)

3)take the equations in the piecewise function and set them equal to each other.

4) plug in x=# for x and solve for b (or a)

now u have b

a=# , b=# and are continuous and differentiable

When x=#, the rate at which (function) is increasing is #a times the rate at which x is increasing. what is the value of a?

1) find y’ of function

2)plug in x=# to y’ function

3)set that equation equal to #a

4)solve for a

Let f(x)= (function). if the rate of change of f at x=a is twice its rate of change at x=# then a=?

1) f’(a)= 2 * f’(#)

2) solve for a