Rules for Differentiation

If f is the function with constant value c,
then
df
d
 (c )  0
dx dx

If n is a positive integer, then
d n
n 1
( x )  nx
dx

If u is a differentiable function of x and c is
a constant, then
d
du
(cu )  c
dx
dx

If u and v are differentiable functions of x,
then their sum and difference are
differentiable at every point where u and v are
differentiable. At such points.
d
du dv
u  v   
dx
dx dx

Find dp/dt.
5
1. p  t  6t  t  16
3
3
2

Do the following curves have any horizontal
tangents? If so, where?
2. y  x  2 x  2
4
2

Do the following curves have any horizontal
tangents? If so, where?
3. y  0.2 x  0.7 x  2 x  5x  4
4
3
2

The product of two differentiable functions u
and v is differentiable, and
d
dv
du
(uv)  u  v
 uv'vu'
dx
dx
dx

Find f’(x).



4. f ( x)  x  1 x  3
2
3

Let y = uv be the product of the functions u
and v. Find y’(2) if
u(2) = 3, u’(2) = -4, v(2) = 1,
and
v’(2) = 2

At a point where v ≠ 0, the quotient y = u/v
of two differentiable functions is
differentiable, and
du
dv
v
u
d u
vu'uv'
dx
dx

 
2
2
dx  v 
v
v

Differentiate.
x 1
6. f x   2
x 1
2

If n is a negative integer and x ≠ 0, then
d n
n 1
( x )  nx
dx

Find an equation for the line tangent to the
following curve at the point (1, 2). Support
your answer graphically.
x 3
7. y 
2x
2
dy
y' 
dx
d2y
y' '  2
dx
4
d
y
4 
y  4
dx
y
d3y
y' ' '  3
dx
n 
n
d y
 n
dx

Find the first four derivatives of the following
function.
8. y  x  5x  2
3
2

Find dy/dx.
1−𝑥
𝑦=
1 + 𝑥2

An orange farmer currently has 200 trees yielding an
average of 15 bushels of oranges per tree. She is
expanding her farm at a rate of 15 trees per year, while
improved husbandry is improving her average annual
yield by 1.2 bushels per tree. What is the current
(instantaneous) rate of increase of her total annual
production of oranges?