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Derivatives: Limit Definition
Related Topics
Wize University Calculus 1 Textbook > Derivatives
The Limit Definition of a Derivative
4 Activities
True or false:
f
(
x
)
=
∣
x
−
3
∣
−
1
f(x) = |x-3|-1
f
(
x
)
=
∣
x
−
3∣
−
1
is differentiable at
x
=
3.
x=3.
x
=
3.
True
False
I don't know
Check Submission
More The Limit Definition of a Derivative Questions:
Limit definition of the derivative (first principles)
If
f
(
x
)
=
1
x
,
f(x)=\frac{1}{\sqrt{x}},
f
(
x
)
=
x
1
,
then find the derivative of the function by first principles.
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f
(
x
)
=
x
+
2
x
−
1
f(x) = x + \sqrt{2x - 1}
f
(
x
)
=
x
+
2
x
−
1
at
x
=
1
x = 1
x
=
1
.
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f
(
x
)
=
x
+
2
x
−
1
f(x) = x + \sqrt{2x - 1}
f
(
x
)
=
x
+
2
x
−
1
at
x
=
1
x = 1
x
=
1
.
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative):
h
(
t
)
=
2
16
−
t
h\left(t\right)=2\sqrt{16-t}
h
(
t
)
=
2
16
−
t
Definition of the Derivative
Carefully state what the definition of the derivative of a function is at the point
x
=
a
x = a
x
=
a
.
Definition of a Derivative
Using the definition of the derivative, find the derivative of
f
(
x
)
=
(
x
+
2
)
2
+
x
f(x)=(x+2)^2+x
f
(
x
)
=
(
x
+
2
)
2
+
x
Definition of a Derivative
Using the definition of the derivative, set up the derivative of
f
(
x
)
=
(
x
+
2
)
2
+
x
f(x)=(x+2)^2+x
f
(
x
)
=
(
x
+
2
)
2
+
x
, but do not solve!
Limit Definition of a Derivative
Find a function
f
(
x
)
f(x)
f
(
x
)
and a number
a
a
a
such that
f
′
(
a
)
=
lim
x
→
0
cos
x
−
1
x
f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}
f
′
(
a
)
=
x
→
0
lim
x
cos
x
−
1
Limit Definition of a Derivative
Given that
f
′
(
2
)
=
4
f^{\prime}(2)=4
f
′
(
2
)
=
4
, find the following limit:
lim
x
→
2
x
−
2
f
(
x
)
−
f
(
2
)
\lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
x
→
2
lim
f
(
x
)
−
f
(
2
)
x
−
2
Limit definition of a derivative
Find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
using the limit definition of the derivative.
f
(
x
)
=
1
x
+
3
for
x
≠
−
3
f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
f
(
x
)
=
x
+
3
1
for
x
=
−
3
Definition of a Derivative
Using the definition of the derivative, find the derivative of
f
(
x
)
=
(
x
+
2
)
2
+
x
f(x)=(x+2)^2+x
f
(
x
)
=
(
x
+
2
)
2
+
x
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f
(
x
)
=
x
+
2
x
−
1
f(x) = x + \sqrt{2x - 1}
f
(
x
)
=
x
+
2
x
−
1
at
x
=
1
x = 1
x
=
1
.
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of
f
(
x
)
=
x
2
f\left(x\right)=\frac{\sqrt{x}}{2}
f
(
x
)
=
2
x
simplifies to
Limit definition of the derivative (first principles)
If
f
(
x
)
=
1
x
,
f(x)=\frac{1}{\sqrt{x}},
f
(
x
)
=
x
1
,
then find the derivative of the function by first principles.
Practice: Definition of Derivative
Practice: Definition of Derivative
For which pair of
f
(
x
)
f\left(x\right)
f
(
x
)
and
a
a
a
does the equality
f
(
a
)
=
lim
x
→
8
x
+
1
−
3
x
−
8
\displaystyle f\left(a\right)=\lim_{x\to8}\ \frac{\sqrt{x+1}-3}{x-8}
f
(
a
)
=
x
→
8
lim
x
−
8
x
+
1
−
3
hold?
Limit Definition of Derivatives
Which of the following limits represents a derivative?
i)
lim
a
→
0
f
(
x
+
a
)
−
f
(
x
)
x
−
a
\displaystyle \lim_{a\to0}\ \frac{f\left(x+a\right)-f\left(x\right)}{x-a}
a
→
0
lim
x
−
a
f
(
x
+
a
)
−
f
(
x
)
ii)
lim
x
→
3
x
−
3
x
−
3
\displaystyle \lim_{x\to3}\ \frac{\sqrt{x-3}}{x-3}
x
→
3
lim
x
−
3
x
−
3
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of
f
(
x
)
=
x
2
f\left(x\right)=\frac{\sqrt{x}}{2}
f
(
x
)
=
2
x
simplifies to
Limit definition of a derivative
Evaluate the following limit:
lim
x
→
0
5
2030
−
(
5
+
x
)
2030
2
x
\displaystyle\lim_{x\rightarrow0} \frac{5^{2030}-(5+x)^{2030}}{2x}
x
→
0
lim
2
x
5
2030
−
(
5
+
x
)
2030
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative):
h
(
t
)
=
2
16
−
t
h\left(t\right)=2\sqrt{16-t}
h
(
t
)
=
2
16
−
t
Q.
\textbf{Q.}
Q.
Consider again the function
f
(
x
)
=
x
2
+
1
f(x)=x^2+1
f
(
x
)
=
x
2
+
1
. What is the instantaneous rate of change of
f
f
f
at
x
=
2
x=2
x
=
2
? Answer using the definition of the derivative.
Q.
\textbf{Q.}
Q.
Show that
f
(
x
)
=
∣
x
−
2
∣
−
1
f(x)=\lvert x-2\rvert-1
f
(
x
)
=
∣
x
−
2
∣
−
1
is not differentiable at
x
=
2
x=2
x
=
2
.
Limit Definition of a Derivative: Working with Given Values
Given that
f
′
(
2
)
=
4
f^{\prime}(2)=4
f
′
(
2
)
=
4
, find
lim
x
→
2
x
−
2
f
(
x
)
−
f
(
2
)
\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
x
→
2
lim
f
(
x
)
−
f
(
2
)
x
−
2
.
Limit Definition of a Derivative
Find a function
f
f
f
and a number
a
a
a
such that:
f
′
(
a
)
=
lim
x
→
0
cos
x
−
1
x
{f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}}
f
′
(
a
)
=
x
→
0
lim
x
cos
x
−
1
Limits: Definition with Fractions
Find the derivative of
g
(
x
)
=
1
2
+
x
\displaystyle g(x)=\frac{1}{2+x}
g
(
x
)
=
2
+
x
1
using the definition.
Limit Definition of a Derivative
Q:
\textbf{Q:}
Q:
Find
f
′
(
2
)
f'(2)
f
′
(
2
)
if
f
(
x
)
=
(
2
x
+
5
)
1
2
f(x)=(2x+5)^{\frac{1}{2}}
f
(
x
)
=
(
2
x
+
5
)
2
1
using the definition of the derivative.
Limit Definition of a Derivative
(Original question is written in long answer. Multiple choice has been used here for convenience)
Let
f
(
x
)
=
x
x
−
3
\displaystyle f(x) = \frac{x}{x - 3}
f
(
x
)
=
x
−
3
x
. Compute
d
f
d
x
\displaystyle \frac{df}{dx}
d
x
df
using the definition of the derivative.
No marks will be given for the use of derivative rules
, but you may use them to check your answer.
Infinite Limits and Continuity
Given the function
f
(
x
)
=
{
arctan
x
+
π
/
2
if
x
≤
0
2
x
if
0
<
x
≤
e
1
ln
x
if
x
>
e
f(x)=\begin{cases} \arctan x+\pi/2&\text{if }x \le 0\\ \frac{2}{x}&\text{if }0<x\le e\\ \frac{1}{\ln x}&\text{if }x>e \end{cases}
f
(
x
)
=
⎩
⎨
⎧
arctan
x
+
π
/2
x
2
l
n
x
1
if
x
≤
0
if
0
<
x
≤
e
if
x
>
e
, which of the following statements is true about
f
(
x
)
f(x)
f
(
x
)
?
Limit Definition of a Derivative
Find a function
f
(
x
)
f(x)
f
(
x
)
and a number
a
a
a
such that
f
′
(
a
)
=
lim
x
→
0
cos
x
−
1
x
f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}
f
′
(
a
)
=
x
→
0
lim
x
cos
x
−
1
Limit Definition of a Derivative
Find the following derivatives using the definition of derivative.
f
(
x
)
=
2
x
3
+
1
f(x)=2x^3+1
f
(
x
)
=
2
x
3
+
1
Find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
using the limit definition of the derivative if
f
(
x
)
=
1
x
2
+
3
\displaystyle f(x)=\frac{1}{x^2+3}
f
(
x
)
=
x
2
+
3
1
Find the following using the definition of derivative.
f
(
x
)
=
x
3
f(x)=\sqrt[3]{x}
f
(
x
)
=
3
x
Find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
using the limit definition of the derivative.
f
(
x
)
=
1
x
+
3
for
x
≠
−
3
f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
f
(
x
)
=
x
+
3
1
for
x
=
−
3
Limit Definition of a Derivative
Evaluate
d
d
x
(
−
2
x
2
−
2
x
+
3
)
\dfrac{d}{dx}(-2x^2-2x+3)
d
x
d
(
−
2
x
2
−
2
x
+
3
)
using the
limit definition of derivative
.
Limit Definition of a Derivative
Evaluate
d
d
x
(
−
x
2
+
7
x
−
2
)
\dfrac{d}{dx}(-x^2+7x-2)
d
x
d
(
−
x
2
+
7
x
−
2
)
using the
limit definition
of derivative.
Limit Definition of a Derivative
Given that
f
′
(
2
)
=
4
f'(2)=4
f
′
(
2
)
=
4
, find
lim
x
→
2
x
−
2
f
(
x
)
−
f
(
2
)
\lim\limits_{x\to 2} \frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
x
→
2
lim
f
(
x
)
−
f
(
2
)
x
−
2
Limit Definition of a Derivative
If
f(x)
is differentiable everywhere, find
A
and
B
.
f
(
x
)
=
{
x
2
+
1
x
≥
0
A
x
+
B
x
<
0
f(x)=\begin{cases}x^2+1 & x \geq 0\\ Ax+B&x<0\end{cases}
f
(
x
)
=
{
x
2
+
1
A
x
+
B
x
≥
0
x
<
0
Limit Definition of a Derivative
Find a function
f
and a number
a
such that
f
′
(
a
)
=
lim
x
→
0
cos
x
−
1
x
f'(a)=\lim\limits_{x\to 0}\frac{\cos x-1}{x}
f
′
(
a
)
=
x
→
0
lim
x
cos
x
−
1
Limit definition of a derivative
Find
f
′
(
2
)
f'(2)
f
′
(
2
)
if
2
f
(
2
)
−
2
f
(
2
+
h
)
h
=
3
h
−
2
\frac{2f(2)-2f(2+h)}{h}=\frac{3}{h-2}
h
2
f
(
2
)
−
2
f
(
2
+
h
)
=
h
−
2
3
for all h near 0.
Practice: Differentiability
Practice: Differentiability
Which of the following is/are
NOT
differentiable at
x
=
0
x=0
x
=
0
?
(Select all that apply)
Find the derivative of
f
(
x
)
=
1
x
+
3
\displaystyle f(x)=\frac{1}{x+3}
f
(
x
)
=
x
+
3
1
, using the limit definition.
Limit Definition of a Derivative
Use the limit definition to find the derivative of
g
(
x
)
=
x
g(x)=\sqrt{x}
g
(
x
)
=
x
.
Limit Definition of a Derivative
Given that
f
′
(
2
)
=
4
f^{\prime}(2)=4
f
′
(
2
)
=
4
, find the following limit:
lim
x
→
2
x
−
2
f
(
x
)
−
f
(
2
)
\lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
x
→
2
lim
f
(
x
)
−
f
(
2
)
x
−
2
Limit Definition of a Derivative
Find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
using the limit definition of the derivative if
f
(
x
)
=
1
x
2
+
3
\displaystyle f(x)=\frac{1}{x^2+3}
f
(
x
)
=
x
2
+
3
1
Limit definition of a derivative
Evaluate the following limit:
lim
x
→
0
5
2030
−
(
5
+
x
)
2030
2
x
\displaystyle\lim_{x\rightarrow0} \frac{5^{2030}-(5+x)^{2030}}{2x}
x
→
0
lim
2
x
5
2030
−
(
5
+
x
)
2030
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative):
h
(
t
)
=
2
16
−
t
h\left(t\right)=2\sqrt{16-t}
h
(
t
)
=
2
16
−
t
Q.
\textbf{Q.}
Q.
Consider again the function
f
(
x
)
=
x
2
+
1
f(x)=x^2+1
f
(
x
)
=
x
2
+
1
. What is the instantaneous rate of change of
f
f
f
at
x
=
2
x=2
x
=
2
? Answer using the definition of the derivative.
Q.
\textbf{Q.}
Q.
Show that
f
(
x
)
=
∣
x
−
2
∣
−
1
f(x)=\lvert x-2\rvert-1
f
(
x
)
=
∣
x
−
2
∣
−
1
is not differentiable at
x
=
2
x=2
x
=
2
.
Limit Definition of a Derivative: Working with Given Values
Given that
f
′
(
2
)
=
4
f^{\prime}(2)=4
f
′
(
2
)
=
4
, find
lim
x
→
2
x
−
2
f
(
x
)
−
f
(
2
)
\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
x
→
2
lim
f
(
x
)
−
f
(
2
)
x
−
2
.
Limit Definition of a Derivative
Find a function
f
f
f
and a number
a
a
a
such that:
f
′
(
a
)
=
lim
x
→
0
cos
x
−
1
x
{f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}}
f
′
(
a
)
=
x
→
0
lim
x
cos
x
−
1
Limit Definition of a Derivative
Q:
\textbf{Q:}
Q:
Find
f
′
(
2
)
f'(2)
f
′
(
2
)
if
f
(
x
)
=
(
2
x
+
5
)
1
2
f(x)=(2x+5)^{\frac{1}{2}}
f
(
x
)
=
(
2
x
+
5
)
2
1
using the definition of the derivative.
Limits: Definition with Fractions
Find the derivative of
g
(
x
)
=
1
2
+
x
\displaystyle g(x)=\frac{1}{2+x}
g
(
x
)
=
2
+
x
1
using the definition.
🦊
TRICKY!
If
f
(
x
)
f(x)
f
(
x
)
is differentiable at
x
=
1
x=1
x
=
1
and
f
′
(
1
)
=
4
f'(1)=4
f
′
(
1
)
=
4
then find the limit
lim
x
→
1
4
x
3
−
4
x
2
f
(
x
)
−
f
(
1
)
\displaystyle \lim_{x\rightarrow1}\frac{4x^3-4x^2}{f(x)-f(1)}
x
→
1
lim
f
(
x
)
−
f
(
1
)
4
x
3
−
4
x
2
.
Limit Definition of a Derivative
The given limit represents the derivative of the function
f
(
x
)
f\left(x\right)
f
(
x
)
at a value
a
a
a
. Identify the function and the value of
a
a
a
.
lim
h
→
0
sin
(
π
2
+
h
)
−
1
h
\displaystyle\lim_{h\ \to0\ }\frac{\sin\left(\frac{\pi}{2}+h\right)-1}{h}
h
→
0
lim
h
sin
(
2
π
+
h
)
−
1
Limit Definition of a Derivative
Find the derivative of
f
(
x
)
=
1
3
x
+
1
\displaystyle f(x) = \frac{1}{3x + 1}
f
(
x
)
=
3
x
+
1
1
by the definition of the derivative.
Calculate the derivative of the function by first principles (i.e. using the limit definition of the derivative)
h
(
t
)
=
2
16
−
t
h\left(t\right)=2\sqrt{16-t}
h
(
t
)
=
2
16
−
t
Find
f
′
(
2
)
if
f
(
x
)
=
(
2
x
+
5
)
1
2
f'(2)\ \text{if}\ f(x)=(2x+5)^{\frac{1}{2}}
f
′
(
2
)
if
f
(
x
)
=
(
2
x
+
5
)
2
1
using the definition of the derivative.
Using the definition of the derivative, find
f
′
(
x
)
f'\left(x\right)
f
′
(
x
)
for
f
(
x
)
=
x
2
−
1
f(x)=x^2−1
f
(
x
)
=
x
2
−
1
.
Limit definition of the derivative (first principles)
If
f
(
x
)
=
1
x
,
f(x)=\frac{1}{\sqrt{x}},
f
(
x
)
=
x
1
,
then find the derivative of the function by first principles.
Limit Definition of a Derivative
Find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
using the limit definition of the derivative if
f
(
x
)
=
2
x
+
3
\displaystyle f(x)=\sqrt{2x+3}
f
(
x
)
=
2
x
+
3
Limit Definition of a Derivative
Evaluate the following limit by recognizing it as a derivative of a function at a certain point.
lim
x
→
1
ln
x
1
−
x
\lim_{x\to1}\ \frac{\ln x}{1-x}
lim
x
→
1
1
−
x
l
n
x
.
Practice: Definition of Derivative
Practice: Definition of Derivative
For which pair of
f
(
x
)
f\left(x\right)
f
(
x
)
and
a
a
a
does the equality
f
(
a
)
=
lim
x
→
8
x
+
1
−
3
x
−
8
\displaystyle f\left(a\right)=\lim_{x\to8}\ \frac{\sqrt{x+1}-3}{x-8}
f
(
a
)
=
x
→
8
lim
x
−
8
x
+
1
−
3
hold?
Limit Definition of Derivatives
Which of the following limits represents a derivative?
i)
lim
a
→
0
f
(
x
+
a
)
−
f
(
x
)
x
−
a
\displaystyle \lim_{a\to0}\ \frac{f\left(x+a\right)-f\left(x\right)}{x-a}
a
→
0
lim
x
−
a
f
(
x
+
a
)
−
f
(
x
)
ii)
lim
x
→
3
x
−
3
x
−
3
\displaystyle \lim_{x\to3}\ \frac{\sqrt{x-3}}{x-3}
x
→
3
lim
x
−
3
x
−
3
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of
f
(
x
)
=
x
2
f\left(x\right)=\frac{\sqrt{x}}{2}
f
(
x
)
=
2
x
simplifies to
Let
f
(
x
)
=
1
4
−
5
x
f(x) = \frac{1}{4 - 5x}
f
(
x
)
=
4
−
5
x
1
. Use the definition of the derivative to compute
f
′
(
3
)
f'(3)
f
′
(
3
)
. No credit will be given for the use of any differentiation rule.
Let
f
(
x
)
=
1
4
−
5
x
f(x) = \frac{1}{4 - 5x}
f
(
x
)
=
4
−
5
x
1
. Use the definition of the derivative to compute
f
′
(
3
)
f'(3)
f
′
(
3
)
. No credit will be given for the use of any differentiation rule.
Limit definition of a derivative
Using the definition of the derivative, solve for the derivative of
f
(
x
)
f(x)
f
(
x
)
at the point
x
=
2
x = 2
x
=
2
:
f
(
x
)
=
1
3
x
+
1
−
1
f(x)= \frac{1}{\sqrt{3x + 1} - 1}
f
(
x
)
=
3
x
+
1
−
1
1
Limit Definition of a Derivative
State the definition of the derivative of a function
f
(
x
)
f(x)
f
(
x
)
at a point
x
=
a
x = a
x
=
a
.
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f
(
x
)
=
x
+
2
x
−
1
f(x) = x + \sqrt{2x - 1}
f
(
x
)
=
x
+
2
x
−
1
at
x
=
1
x = 1
x
=
1
.
Definition of the Derivative
Carefully state what the definition of the derivative of a function is at the point
x
=
a
x = a
x
=
a
.
Definition of a Derivative
Using the definition of the derivative, find the derivative of
f
(
x
)
=
(
x
+
2
)
2
+
x
f(x)=(x+2)^2+x
f
(
x
)
=
(
x
+
2
)
2
+
x
Limit definition of a derivative
Find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
using the limit definition of the derivative.
f
(
x
)
=
1
x
+
3
for
x
≠
−
3
f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
f
(
x
)
=
x
+
3
1
for
x
=
−
3
Find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
using the limit definition of the derivative if
f
(
x
)
=
1
x
2
+
3
\displaystyle f(x)=\frac{1}{x^2+3}
f
(
x
)
=
x
2
+
3
1