MATH1013 Tutorial 6

MATH1013 Tutorial 6
Two Special Limits
eh − 1
= 1 since e is defined to be the number such that the limit equals 1.
lim
h→0
h
That is, the only number such that y = ax has derivative 1 at x = 0.
sin x
= 1 by considering the following graph and comparing the areas of the circular sector
lim
x→0 x
sin x
and the two other triangles for positive x. The left hand limit is equal since
is even.
x
1
x
1
9h − 3h
.
h→0
h
1. Evaluate lim
2. Evaluate lim
tan 3x
.
sin 8x
3. Evaluate lim
1 − cos x
.
x2
x→0
x→0
Derivatives of Trigonometric Functions
The derivatives of the six trigonometric functions are summarized as follows:
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec2 x
dx
d
cot x = − csc2 x
dx
d
sec x = sec x tan x
dx
d
csc x = − csc x cot x
dx
We can obtain the derivatives of the co-functions by adding negative signs and switching each corresponding term
from the derivative of the three trigonometric functions.
Prepared by Leung Ho Ming
Homepage: http://ihome.ust.hk/~malhm
1
The Chain Rule
If g is differentiable at x and f is differentiable at g(x), then the composite function F = f ◦ g is differentiable at x
and
F 0 (x) = f 0 (g(x)) · g 0 (x).
In Leibniz notation, if y = f (u) and g = u(x) are differentiable functions then
dy du
dy
=
·
.
dx
du dx
4. Find the derivatives of the following functions.
(a) f (x) = (x2 + sec x + 1)3
(b) g(x) = sin x2 cos 2x
(c) h(x) = 4 sin
(d) p(x) =
p
1+
√
x
cos x
1 + sin x
5. Suppose that the functions f and g and their derivatives have the following values at x = 0 and x = 1
x
0
1
f (x)
1
3
g(x)
1
−4
f 0 (x)
5
−1/3
g 0 (x)
1/3
−8/3
Find the derivatives with respect to x of the following combinations at x = 0
(a) f (x)(g(x))3
(b) f (g(x))
(c) g(f (x))
(d) f (x + g(x))
2
Implicit Differentiation
Suppose the relation between x and y is given as f (x, y) = 0, for example x2 + y 2 − r2 = 0. To obtain the derivative
dy/dx, we simply differentiate both sides by treating y as a function of x.
With implicit differentiation, we can obtain the derivatives of the six inverse trigonometric functions:
d
1
sin−1 x = √
dx
1 − x2
d
1
cos−1 x = − √
dx
1 − x2
d
1
tan−1 x =
dx
1 + x2
d
1
cot−1 x = −
dx
1 + x2
d
1
√
sec−1 x =
dx
|x| x2 − 1
d
1
csc−1 x = − √
dx
|x| x2 − 1
Indeed only the derivatives of sin−1 x and tan−1 x are important, and are suggested to be memorized.
6. Show that
d
1
√
sec−1 x =
for |x| > 1 by implicitly differenting y = sec−1 x, or sec y = x.
dx
|x| x2 − 1
x2
y2
7. An ellipse centered at origin that passes through (−a, 0), (a, 0), (0, −b) and (0, b) has equation 2 + 2 = 1 where
a
b
x0 x y0 y
a, b > 0. Show that the equation of tangent line at (x0 , y0 ) is 2 + 2 = 1.
a
b
8. Suppose f is a one-to-one differentiable function and its inverse f −1 is also differentiable.
(a) Differentiate f ◦ f −1 (x) = x implicitly to show that (f −1 )0 (x) =
(b) If f (4) = 5 and f 0 (4) = 32 , find (f −1 )0 (5).
3
1
.
f 0 (f −1 (x))
Derivatives of Exponential and Logarithmic Functions
The derivative of ex is ex . If the exponent is not in base e, then
d x
d x ln a
a =
e
= ex ln a · ln a = ax ln a.
dx
dx
d
1
ln x = . If the logarithm is not in base e, then
dx
x
The derivative of ln x is
d ln x
1
d
loga x =
=
.
dx
dx ln a
x ln a
If f (x) is a complicted function involving products, quotients or powers, we may apply logarithms:
y = f (x) ⇒ ln y = ln f (x)
and differentiate implicitly to obtain f 0 (x).
9. Find the derivatives of the functions.
5
(x + 1)(x − 1)
where x > 2.
(a) f (x) =
(x − 2)(x + 3)
(b) g(x) = (x + 2)x+2
√
(c) h(x) = (sin x)
x
4