d/dx (sinx)= cosx     read as "the derivative of y=sinx is y=cosx"
and its RULE is: If y=sin f(x), then y'= cosf(x) x f'(x)

d/dx (cosx)= -sinx     read as "the derivative of y=cosx is y=-sinx"
and its RULE is: If y=cos f(x), then y'= -sinf(x) x f'(x)

*remember to use derivative rules when differentiating!

Real Life Applications:
The graphs are used to measure:
-music (instrument vibrations and acoustics)
-ocean tides and currents

Procedure for Sample Problems:
Example: Find the Derivative
y= 2cos4x
Steps: Take the derivative of y=cosx, which is y'=-sinx. Take the derivative of 4x, which is 4. Multiply it by the coefficient in the front.
y'=-2sin4x (4)
y'=-8sin4x

Example: Differentiate the function
y=3sinx cosx
Steps: Here is where you will use the product rule. Find the derivatives of both parts of the function. Factor, and then apply trigonometric identities if possible.
y'=(3sinx)(-sinx) + (cosx)(3cosx)
y'=-3sin²x + 3cos²x
y'=3(cos²x - sin²x)
y'=3cos(2x)

d/dx (tanx)= sec²x     read as "the derivative of y=tanx is y=sec²x
and its RULE is: if y=tan f(x), then y'= sec² f(x) x f'(x)

Real Life Applications:
The graph is used to measure:
-electrical currents
-GPS/Satellite Orbital Mechanics

Procedure for Sample Problems:
Example: Find the Derivative
y=tan6x
Steps: Take the derivative of y=tanx, which is y'=sec²x. Then take the derivative of 6x which is 6.
y'=sec²6x (6)
y'=6sec²6x

Example: Find the Derivative:
y=tan(cosx)
Steps: Take the derivatives of y=tanx and y=cosx, which are y'=sec²x and y'=-sinx. Multiply the equation by the derivative of what is in the bracket, y'=-sinx.
y'=sec²x(cosx) (-sinx)
y'=-sinx sec²x (cosx)

-Knowing how to find derivatives of exponential functions allows us to compare rates of change of functions describing radioactive decay, population growth or loan interest rates.