The Derivative of the basic Trigonometric Functions
Sketch the derivative of y = sinx. Do you recognize it?
Using first principles you can derive this…
Sketch the derivative of y = cosx. Do you recognize it?
Sketch the derivative of y = tanx. Do you recognize it?
You could also have obtained this derivative by first principles or by differentiation using
sin x
the quotient rule and the fact that tan x = cos x
Sketch the derivative of y = cot x. Do you recognize it?
You could also have obtained it by first principles or by using the quotient rule to
cos x
1
differentiate: y =
or by differentiating y =
sin x
tan x
Sketch the derivative of y = secx. Do you recognize it?
The derivative of the secant and cosecant graphs are not easily recognizable. You can
determine them by using trigonometric identities and differentiating.
Example #1
Differentiate each of the following:
a)
y = sin( 4 x )
b)
y = sin( x − 3)
c)
y = sin( x 3 )
d)
y = sin 4 x
e)
y = cos( 3 x + 2)
f)
y = 3 tan( 2 x)
Example #2
Differentiate:
y = x 2 cos(3 x )
Example #3
If sin x + sin y = 1, find the derivative of y with respect to x.
Example #4
Differentiate: f ( x) =
1
1 + tan x
Example #5
3
2
Differentiate: y = 2 csc (3 x )
Example #6
2
If tan y = x , fine the derivative of y with respect to x. When y =
π
4
Example #7
2
Find the derivative of : y = tan (cos(5 x ))
Example #8
The top of a ladder 3m long is slipping down a wall at a rate of 0.2m/minute. How
quickly is the angle that the base of the ladder makes with the ground changing when it is
thirty degrees?