3.3 Derivatives of Trig functions
3.3 Derivatives of Trig functions
Differentiating polynomials is (relatively) easy. Unfortunately trigonometric functions require some
more work:
f (x) = sin x
Differentiating Sine The first thing to do is look at the
1
picture. For reasons to be seen in a moment, we work
in radians. We may visually differentiate the graph of f (x) = sin x: think about where the curve is:
2 x
horizontal = f (x) = 0
-1
increasing = f (x) > 0
f (x)
decreasing = f (x) < 0
f (x) certainly looks very like a cosine curve, but notice
that we have no scale on the y-axis. We should suspect
that
d dx
sin x
=
cos x, but we still need a proof.
2 x
Return to the definition of derivative. To differentiate f (x) = sin x, we need to evaluate the limit as h 0 of the following difference quotient:
f (x
+
h) h
-
f (x)
=
sin(x
+
h) h
-
sin
x
=
sin x
cos h
+
cos h
x
sin
h
-
sin
x
=
sin
x
cos
h h
-
1
+
cos
x
sin h
h
If our guess above is correct, then we need to prove the following:
Theorem. If angles are measured in radians, then
lim
h0
sin h h
=
1
and
lim
h0
cos
h- h
1
=
0
It follows that
d dx
sin x
=
cos x.
Proof. Everything follows from the picture and the Squeeze Theorem.
Consider a segment of a circle of radius 1 and angle h > 0.
By the definition of radians, the arc-length of the segment is
also h. Drawing the triangles as shown, it should be clear
that all lengths are as claimed.
It is immediate that Now recall that the
asrineaho ................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- section 9 1 introduction to di erential equations
- unit 1 differentiation mit opencourseware
- integration of trigonometric functions
- exam2 penn math
- 3 3 derivatives of trig functions
- section 7 3 some trigonometric integrals
- 8 2 trigonometric integrals chapter 8 techniques of
- 8 4 trigonometric integrals chapter 8 techniques of
- review for midterm i
- chapter 7 successive differentiation
Related searches
- derivatives of trig functions definition
- derivatives of trig functions
- inverse derivatives of trig functions
- derivatives of inverse functions pdf
- derivatives of trig functions examples
- derivatives of trig identities
- derivatives of trigonometric functions pdf
- derivatives of trigonometric functions worksheet
- derivatives of trig function
- derivatives of trig functions rules
- derivatives of trigonometric functions list
- all derivatives of trig functions