Math 103 – Rimmer
3.3 Derivative Formulas
Let f ( x ) and g ( x ) be differentiable functions and let k be a constant.
1. Constant function
y=k
3. Constant Multiple Rule
y = k f ( x)
2. Linear function
y = mx + b
from last class
4. The Power Rule
ex : y =
n
y = x , n any real number
1
= x −1/ 2
x
ex : y = x 4
5. The Sum Rule
y = f ( x) + g ( x )
7. The Product Rule
y = f ( x) g ( x)
6. The Difference Rule
y = f ( x) − g ( x)
Math 103 – Rimmer
3.3 Derivative Formulas
from last class
ex : y = 4 x − 7 x 2
y′ ≠ f ′ ( x ) g ′ ( x )
ex : Let f ( x ) = x ⋅ g ( x ) , g ( 4 ) = 8, and g ′ ( 4 ) = 7. Find f ′ ( 4 ) .
Math 103 – Rimmer
3.3 Derivative Formulas
8. The Quotient Rule
y=
f ( x)
g ( x)
ex : Let y =
y′ ≠
f ′( x)
g′( x)
3x + 1
. Find y′ (1) .
x2 + 1
ex : Find the equation of the tangent line to y =
Math 103 – Rimmer
3.3 Derivative Formulas
3x + 1
at (1, 2 ) .
x2 + 1
The normal line to a curve at a point P is the line
through P that is ___________________________ .
ex : Find the equation of the normal line to y =
3x + 1
at (1, 2 ) .
x2 + 1
⊥ lines have
1)
2)
Math 103 – Rimmer
3.3 Derivative Formulas
ex : Let Q ( x ) =
F ( x)
where F and G are the functions whose graphs are shown. Find Q′ ( 7 ) .
G ( x)
1
4
−2
3
Math 103 – Rimmer
3.3 Derivative Formulas
ex : Find the points on the curve y = 2 x3 + 3 x 2 − 12 x + 1 where the tangent line is horizontal.
ex : Find the equation of the tangent lines to the curve
y=
Math 103 – Rimmer
3.3 Derivative Formulas
x −1
that are parallel to the line x − 2 y = 2.
x +1
Math 103 – Rimmer
3.3 Derivative Formulas
ex : Find the value of c such that the line y =
3
x + 6 is tangent to the curve y = c x .
2
ex : Let
if x ≤ 1
 2− x
f ( x) =  2
 x − 2 x + 2 if x > 1
Is f differentiable at 1? Sketch the graphs of f and f ′.
In order for f to be differentiable at 1 it must be ____________.
Is f continuous at 1? If the answer is no, then __________.
lim− f ( x ) =
x →1
lim+ f ( x ) =
x →1
In order for f to be differentiable, the slope of the tangent line from
the left _______________________________________________
Math 103 – Rimmer
3.3 Derivative Formulas