Download Differentiation Rules

Differentiation Rules
MATH 150 Spring 2012
1/1
Power function
From the definition of the derivative it is easy to see that
d
(c) = 0
dx
for every constant c ∈ R.
Next,
d
(x) = 1
dx
and more generally
d n
(x ) = nx n−1
dx
for every n ∈ N.
Fact
If α ∈ R then
d α
(x ) = αx α−1
dx
2/1
Power function
Example
Differentiate
(a) f (x) =
1
x2
(b) f (x) =
√
3
x2
Example
√
Find an equation of the tangent line to the curve y = x x at the point
(1, 1).
3/1
Derivative of the sum
Fact
If c ∈ R, and f and g are both differentiable, then
(a)
d
dx [cf
(b)
d
dx [f
d
(x)] = c dx
f (x)
(x) ± g (x)] =
d
dx f
(x) ±
d
dx g (x)
Example
Differentiate
f (x) = x 7 − 6x 5 + 3x 4 − x 2 +
√
x
4/1
Derivative of the sum
Example
Let
f (x) =
x2
mx + b
if x 6 2
if x > 2
Find the constants m and b which make f (x) differentiable everywhere.
Example
Let
f (x) =
2−x
x 2 − 2x + 2
if x 6 1
if x > 1
Where is f (x) differentiable?
5/1
Exponential function
Observe that if f (x) = ax then
ax+h − ax
ah − 1
= ax · lim
= ax · f ′ (0) = f ′ (0) · f (x)
h→0
h→0
h
h
f ′ (x) = lim
Here, f ′ (0) is the slope of the curve y = ax at the point (0, 1).
Fact
limh→0
ah −1
h
= 1 if and only if a = e = 2.718281828 . . ., hence,
d x
(e ) = e x
dx
Example
At what point on the curve y = 1 + 2e x − 3x is the tangent line parallel
to the line 3x − y = 5 ?
6/1
The product and quotient rules
Fact
If f and g are both differentiable, then
d
d
d
[f (x) · g (x)] =
f (x) · g (x) + f (x) ·
g (x)
dx
dx
dx
Example
Differentiate
√
f (x) = ( x + 3)(x 2 − 5x)
7/1
The product and quotient rules
Fact
If f and g are both differentiable, then
d
f (x) · g (x) − f (x) ·
d f (x)
= dx
dx g (x)
g (x)2
d
dx g (x)
Example
Differentiate
f (x) =
x2 + x − 2
x3 + 6
8/1
The product and quotient rules
Example
Differentiate
√
( x + 3)(x − 1)
f (x) =
x 2 − 5x
Example
Find an equation of the tangent line to the curve y =
(0, 1).
ex
1−x
at the point
9/1
Derivatives of trigonometric functions
Consider f (x) = sin x. Let us compute f ′ (x). By definition we have
f ′ (x) = lim
h→0
f (x + h) − f (x)
sin(x + h) − sin x
= lim
h→0
h
h
sin x · cos h + cos x · sin h − sin x
= lim
h→0
h
sin x · cos h − sin x
cos x · sin h
= lim
+
h→0
h
h
cos h − 1
sin h
= lim sin x
+ cos x
h→0
h
h
= sin x · lim
h→0
cos h − 1
sin h
+ cos x · lim
h→0 h
h
10 / 1
Derivatives of trigonometric functions
Fact
limh→0
sin h
h
=1
From the fact above we can derive that
lim
h→0
cos h − 1
=0
h
Indeed,
cos h − 1
cos h − 1 cos h + 1
cos2 h − 1
lim
= lim
·
= lim
h→0
h→0
h→0 h · (cos h + 1)
h
h
cos h + 1
− sin2 h
sin h
sin h
= lim
= − lim
·
h→0 h · (cos h + 1)
h→0
h
cos h + 1
sin h
sin h
0
= − lim
· lim
= −1 ·
=0
h→0 h
h→0 cos h + 1
1+1
11 / 1
Derivatives of trigonometric functions
Finally, returning to the derivative of f (x) = sin x we get
f ′ (x) = sin x · lim
h→0
cos h − 1
sin h
+ cos x · lim
h→0 h
h
= sin x · 0 + cos x · 1 = cos x
Fact
(a)
(b)
d
dx (sin x) = cos x
d
dx (cos x) = − sin x
Example
Differentiate
f (x) =
1 + sin x
x + cos x
12 / 1
Derivatives of trigonometric functions
Example
Find the points on the curve y =
horizontal.
cos x
2+sin x ,
where the tangent line is
Example
Compute
lim
x→0
sin 4x
sin 6x
Example
Compute
lim
x→1
sin(x − 1)
x2 + x − 2
13 / 1
Derivatives of trigonometric functions
Next,
d
d
(tan x) =
dx
dx
sin x
cos x
=
cos x ·
d
dx (sin x)
− sin x ·
cos2 x
d
dx (cos x)
cos x · cos x − sin x · (− sin x)
cos2 x + sin2 x
1
=
=
= sec2 x
cos2 x
cos2 x
cos2 x
Similarly we get
d
(cot x) = − csc2 x
dx
=
Fact
d
d
dx (sin x) = cos x,
dx (cos x) = − sin x
d
d
2
2
dx (tan x) = sec x,
dx (cot x) = − csc x
d
d
dx (csc x) = − csc x · cot x,
dx (sec x) = sec x
· tan x
14 / 1
Derivatives of trigonometric functions
Example
Differentiate
f (x) =
sec x
1 + tan x
Example
Find the 27th derivative of cos x
15 / 1