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By Gökhan Bilhan
Calculus 1
(Week 8)-More About Derivatives and L'Hospital's Rule
Here is inverse trigonometric functions:
y = arcsinx, then let's nd y ′ .
Example
If y = arctanx, then let's nd y ′ .
Example
If y = arctan(x2 ) − (arctanx)2 , then let's nd y ′ .
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By Gökhan Bilhan
Example
If y = arcsin(sinx), then let's nd y ′ .
Higher Order Derivatives
d2 f
represents the second derivative of f .
dx2
d3 f
If y = f (x), then y ′′′ = D(D2 (f )) = D3 f = 3 represents the third derivative of f .
dx
If y = f (x), then y ′′ = D(D(f )) = D2 f =
In general, If y = f (x), then y (n) = Dn f =
Examples
√
1-) Let's nd y ′′ , if y = (1 + x)3
dn f
represents the n−th derivative of f .
dxn
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By Gökhan Bilhan
d8
x2
8!
1-) Let's show that 8 (
)=
.
dx 1 − x
(x − 1)9
Exercises
1. If y = arccosx, then let's nd y ′ .
2. If y = arccotx, then let's nd y ′ .
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By Gökhan Bilhan
√
√
3. If y = 3 arcsinx − arcsin 3 x, then let's nd y ′ .
4. If y = arc(cos(cos(π 2 + 1))), then let's nd y ′ .
5. Find y ′′ , if y =
x
1+x
6. Find y (6) as fast as possible, if y = x(2x − 1)2 (x + 3)3
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By Gökhan Bilhan
L'Hospital's Rule
Let f and g be two dierentiable functions on [a, b) with g ′ (x) ̸= 0 for every x ∈ [a, b). If for a
b where either b is a real number or b = ±∞
1-) limx→b− f (x) = limx→b− g(x) = 0 ( or ∞ ) and
2-) limx→b−
f ′ (x)
= L (L is possibly ±∞ )
g ′ (x)
then
limx→b−
Example
Example
f (x)
= L too.
g(x)
(Negative Example) Evaluate limx→0
Evaluate limx→∞
x − sinx
x
1
x
sinx
x2 sin
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By Gökhan Bilhan
secx + 1
tanx
Example
Evaluate lim
Example
Evaluate limx→0+ ( − cotx)
Example
Evaluate limx→∞ (x) x
π
x→
2
1
x
1
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By Gökhan Bilhan
Example
Evaluate limx→∞ ( x1 )sin x
1
Exercises
ex
1. Evaluate limx→1 2
x
2. Evaluate limx→∞
3. Evaluate limx→0
lnx
x−1
tanx − x
x3
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By Gökhan Bilhan
4. Evaluate limx→π−
sinx
1 − cosx
5. Evaluate limx→0+ xlnx
6. Evaluate limx→ π2 − secx − tanx
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By Gökhan Bilhan
(Week 8)-More About Derivatives and L'Hospital's Rule
Exercises
1. Evaluate limx→0+ (1 + sin4x)cosx
2. Evaluate limx→0+ xx
3. Evaluate limx→0
tanpx
, where p, q are real numbers.
tanqx
4. Evaluate limx→∞
ex
x3
10
By Gökhan Bilhan
1
x
5. Evaluate limx→0 ( − cosecx)
6. Evaluate limx→0+ − (lnx)x
7. Evaluate limx→0
arcsinx
x
8. Evaluate limx→( π2 )− sec7xcos3x
9. Evaluate limx→1
xa − 1
, where a, b are real numbers.
xb − 1
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Calculus 1