Download Derivatives of Sinusoidal Functions

derivatives of trigonometric, exponential & logarithmic functions
derivatives of trigonometric, exponential & logarithmic functions
Derivative of f (x) = sin x
MCV4U: Calculus & Vectors
While we have dealt with derivatives of polynomial, radical
and reciprocal functions, we have not yet dealt with the
derivatives of sinusoidal functions like f (x) = sin x.
Derivatives of Sinusoidal Functions
We might calculate the rate of change every
compare it to the graph of sine.
π
3
radians and
(Sine and Cosine)
J. Garvin
From the graph, it appears that the rate of change of the
sine function is the cosine function.
J. Garvin — Derivatives of Sinusoidal Functions
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Slide 1/17
derivatives of trigonometric, exponential & logarithmic functions
derivatives of trigonometric, exponential & logarithmic functions
Derivative of f (x) = sin x
Derivative of f (x) = sin x
To establish the derivative of the sine function, use the limit
definition of the derivative.
Use the multiplicative property of limits.
sin(x + h) − sin x
h→0
h
f 0 (x) = lim
Use the angle sum identity for sine.
f 0 (x) = lim
h→0
sin x cos h + sin h cos x − sin x
h
Common factor sin x.
sin x(cos h − 1) + sin h cos x
f 0 (x) = lim
h→0
h
J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
f 0 (x) = lim
h→0
sin x(cos h − 1)
sin h cos x
+ lim
h→0
h
h
Since we are concerned with the limit of h, both sin x and
cos x can be treated like constants outside of the limits.
f 0 (x) = sin x · lim
h→0
sin h
cos h − 1
+ cos x · lim
h→0 h
h
At this point, we need to establish values for the two limits
above.
Proofs of these limits generally rely on the Squeeze Theorem
and some trigonometric inequalities, but some informal
arguments should convince us of their values.
J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
Derivative of f (x) = sin x
Derivative of f (x) = sin x
Recall that in radian measure, arc length of a circle is given
by a = r θ, where θ is the angle subtending the arc and r is
the radius of the circle.
Now consider what happens as θ → 0.
Consider the unit circle below. When r = 1, a = θ. The
height of a right triangle inside of the sector has a height of
sin θ.
The length of the arc, θ, and the height of the triangle, sin θ,
become closer in value. If θ was infinitesimally small, then θ
and sin θ would essentially have the same value.
sin θ
sin θ
Their ratio,
, would be 1. Therefore, lim
= 1.
θ→0 θ
θ
J. Garvin — Derivatives of Sinusoidal Functions
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J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
derivatives of trigonometric, exponential & logarithmic functions
Derivative of f (x) = sin x
Derivative of f (x) = sin x
cos θ − 1
A similar argument can be used for lim
.
θ→0
θ
Consider what happens to the horizontal component of a
right triangle, cos θ, as θ → 0.
Now return to the previous definition of the derivative of
f (x) = sin x and substitute these values.
sin h
cos h − 1
+ cos x · lim
h→0 h
h
= sin x · 0 + cos x · 1
f 0 (x) = sin x · lim
h→0
= cos x
This confirms our graph earlier.
Derivative of the Sine Function
If θ was infinitesimally small, then cos θ would essentially
have the same length as the radius, 1.
cos θ − 1
Therefore, cos θ − 1 = 1 − 1 = 0, and lim
= 0.
θ→0
θ
J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
If f (x) = sin x, then f 0 (x) = cos x.
J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
Derivative of f (x) = cos x
Derivatives Involving Sinusoidal Functions
Use the same process for the derivative of f (x) = cos x.
We can use the derivative rules developed earlier to find the
derivatives of functions involving either sine or cosine.
cos(x + h) − cos x
f 0 (x) = lim
h→0
h
cos x cos h − sin x sin h − cos x
= lim
h→0
h
cos x(cos h − 1) − sin x sin h
= lim
h→0
h
cos h − 1
sin h
= cos x lim
− sin x lim
h→0
h→0 h
h
= cos x · 0 − sin x · 1
= − sin x
Derivative of the Cosine Function
If f (x) = cos x, then f 0 (x) = − sin x.
J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
Example
Determine the derivative of f (x) = 10 cos x + 4.
Since 10 is a constant multiple, and 4 is a constant,
f 0 (x) = −10 sin x.
Example
Determine the derivative of f (x) = sin x − cos x.
Using the difference rule for derivatives,
f 0 (x) = cos x − (− sin x) = cos x + sin x.
J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
Derivatives Involving Sinusoidal Functions
Derivatives Involving Sinusoidal Functions
Example
Example
Determine the derivative of y = cos(2x 3 + 5x).
Determine the derivative of y = 2 sin x cos x.
Use the chain rule, where the inner function is u = 2x 3 + 5x
and the outer function is y = cos u.
Use the product rule to differentiate.
dy
dx
= − sin(2x 3 + 5x)(6x 2 + 5)
dy
dx
= 2(cos x cos x − sin x sin x)
= 2(cos2 x − sin2 x)
= 2 cos 2x
Example
Determine the derivative of y = 7 sin3 x + 2 cos2 x.
Use the chain rule and the fact that sinn x = (sin x)n .
dy
dx
= 21(sin x)2 (cos x) + 4(cos x)(− sin x)
= 21 sin2 x cos x − 4 sin x cos x
J. Garvin — Derivatives of Sinusoidal Functions
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An alternative solution uses the chain rule and the identity
y = 2 sin x cos x = sin 2x.
dy
dx
= cos(2x)(2)
= 2 cos 2x
J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
derivatives of trigonometric, exponential & logarithmic functions
Derivatives Involving Sinusoidal Functions
Derivatives Involving Sinusoidal Functions
Example
Example
Determine the slope of the tangent to f (x) = 2x cos x when
x = 5π
3 .
Find any values of x for which the tangent to y = 3 sin2 x, on
the domain [0, 2π], has a slope of 32 .
Use the product rule to differentiate.
Using the chain rule, the derivative is dy
dx = 6 sin x cos x, or
dy
dx = 3 sin 2x after applying the double-angle formula.
f 0 (x) = 2 cos x + 2x(− sin x)
Find f 0
5π
3
Set
= 2(cos x − x sin x)
dy
dx
=
3
2
and solve for 2x.
3 sin 2x =
for the slope of the tangent.
√ 1
5π
f 0 5π
− 23
3 =2 2 − 3
=1+
sin 2x =
3
2
1
2
2x = sin−1
√
5 3π
3
2x =
π
6
1
2
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J. Garvin — Derivatives of Sinusoidal Functions
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derivatives of trigonometric, exponential & logarithmic functions
derivatives of trigonometric, exponential & logarithmic functions
Derivatives Involving Sinusoidal Functions
Derivatives Involving Sinusoidal Functions
Since sin 2x has a period of π, two cycles will be completed
on the interval [0, 2π]. Thus, there should be four values of x
for which this equation is true.
A graph of y and
dy
dx
is below.
The first two can be found by using 2x as the reference angle.
2x =
x=
π
6
π − 2x =
π
12
x=
=
π
6
π−
2
π
6
5π
12
Find the other two by adding the period, π, to each value.
x =π+
=
π
12
13π
12
x =π+
=
5π
12
17π
12
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derivatives of trigonometric, exponential & logarithmic functions
Questions?
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