Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
x
2
f
x
8
g x
2
f  x
1/3
g x
–3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(a) 2 f
d
dx
x
 2 f
3
–4
at x = 2
At x = 2:
 x   
2 f  x
1 2
2 f 2  2   
3 3
Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
x
2
f
x
g x
8
2
f  x
1/3
g x
–3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(b) f
d
dx
–4
3
x  g x
at x = 3
 f  x   g  x  
 f  x  g x
At x = 3:
f   3   g   3   2  5
Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
x
2
f
x
g x
8
2
f  x
1/3
g x
–3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(c) f
d
dx
x
 f
3
–4
g  x  at x = 3
x
g  x    f
 x g x  g  x
At x = 3:
f  x
f  3  g   3   g  3  f   3    3   5     4   2
 1 5  8

Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
x
2
f
x
g x
8
f  x
2
g x
–3
1/3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
–4
3
g  x  at x = 2
g 2 f 2  f 2 g2
d
 f  2  g  2   
2
dx
 g  2  
(d) f
x

2 1 3   8   3 
2
2

74 3
4

37
6
Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
x
2
f
x
8
g x
2
f  x
1/3
g x
–3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(e) f
d
dx
 g  x 
3
–4
at x = 2
f  g 2  f  g  2  g  2   f  2 g  2
1
    3  1
3
Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
x
f
2
x
g x
8
2
f  x
g x
–3
1/3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(f)
f
d
dx
x
–4
3
at x = 2
1
f 2 
2
d
f  2  dx
f 2
f 2 
2

f 2
1 3
2 8

1
12 2
Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
f
x
2
x
8
g x
f  x
2
g x
–3
1/3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(g) 1 g
d
dx
2
x
3
–4
at x = 3
2
 g  3     2  g  3  
3
d
dx
g 3  

2 g 3
 g  3  
2 5
 4 
3

5
32
3
Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
f
x
2
x
g x
8
2
f  x
1/3
g x
–3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(h)
d
dx
f
2
f
 x  g x
2
2
2
at x = 2
2  g 2
2
d
d


2 f 2
f 2  2g 2
g  2 

2
2
dx
dx

f 2  g 2 
1

–4
3
Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
f
x
2
x
g x
8
f  x
2
1/3
g x
–3
5
2
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
3
(h)

 x   g  x  at x = 2
2 f 2  g 2 g2
f
f
–4
3
2
2
f
2
2  g 2

 8  1 3    2    3 
8 2
2

2
 10 3
68

 10 3
2 17

2
5
3 17
Slopes of Parametrized Curves
A parametrized curve (x(t), y(t)) is differentiable at t if x and y
are differentiable at t. At a point on a differentiable parametrized
curve where y is also a differentiable function of x, the
derivatives dy/dt, dx/dt, and dy/dx are related by the Chain Rule:
dy
dt

dy dx
dx dt
Usually, we write this in a different form…
If all three derivatives exist and d x d t  0 ,
dy
dx

dy dt
dx dt
Practice Problems
Find the equation of the line tangent to the curve at the point
defined by the given value of t.
x  sin 2  t
y  cos 2  t
t  1 6
Find the three derivatives:
dx
dt
dy
dt
dy
dx
  cos 2  t 
d
dt
   sin 2  t 

dy dt
dx dt

 2 t 
d
dt
 2  co s 2  t
 2 t 
  2  sin 2  t
 2  sin 2  t
2  cos 2  t
  tan 2  t
Practice Problems
Find the equation of the line tangent to the curve at the point
defined by the given value of t.
x  sin 2  t
y  cos 2  t
t  1 6
2
2  
3 1

The line passes through: sin
, cos
, 

   
6
6  
2 2 

2
And has slope:  tan
 3
6
Equation of the tangent line:
y
1
2


3
3x


2 

y
3x  2
Practice Problems
Find the equation of the line tangent to the curve at the point
defined by the given value of t.
x  2t  3
yt
2
Derivatives:
dx
dt

 4t
dy
 4t
dt
Point: 2   1   3,   1 
2
4
3
t  1
dy
dx
4
   5,1 

dy dt
dx dt

t
4t
Slope:   1   1
2
Equation of the tangent line:
y  1  1 x  5
4t
3
y  x4
2