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Differentiation laws
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Differentiation Laws
1 Activity
What is the derivative of the function
f
(
x
)
=
x
3
+
1
x
2
f(x)=\frac{x^3+1}{x^2}
f
(
x
)
=
x
2
x
3
+
1
?
3
2
x
\frac{3}{2}x
2
3
x
1
−
2
x
1-\frac{2}{x}
1
−
x
2
1
−
2
x
3
1-\frac{2}{x^3}
1
−
x
3
2
2
x
3
−
1
\frac{2}{x^3}-1
x
3
2
−
1
None of the above
I don't know
Check Submission
More Differentiation Laws Questions:
Derivative of Polynomials: Product Rule
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
Differentiation laws
What is the derivative of the function
f
(
x
)
=
x
3
+
1
x
2
f(x)=\frac{x^3+1}{x^2}
f
(
x
)
=
x
2
x
3
+
1
?
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere. Answers are in the form
(
A
,
B
)
\left(A,B\right)
(
A
,
B
)
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Derivative of Polynomials: Product Rule
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Derivatives: Logarithmic Functions
Compute the derivative of
f
(
x
)
=
x
x
+
1
f(x) = x^{x + 1}
f
(
x
)
=
x
x
+
1
. Remember that
log
x
=
log
e
x
=
ln
x
\log x = \log_e x = \ln x
lo
g
x
=
lo
g
e
x
=
ln
x
.
The chain rule
Find the derivative of
h
(
x
)
=
log
(
cos
(
x
)
)
h(x) = \log(\cos(x))
h
(
x
)
=
lo
g
(
cos
(
x
))
. Remember that
log
x
=
log
e
x
=
ln
x
\log x = \log_e x = \ln x
lo
g
x
=
lo
g
e
x
=
ln
x
.
The Quotient Rule
Find the derivative of
g
(
x
)
=
x
2
−
5
2
x
+
1
g(x) = \frac{x^2 - 5}{2x + 1}
g
(
x
)
=
2
x
+
1
x
2
−
5
The product rule
Find the derivative of
f
(
x
)
=
x
3
e
x
f(x) = x^3 e^x
f
(
x
)
=
x
3
e
x
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Derivative of Polynomials: Product Rule
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
Is
f
(
x
)
f(x)
f
(
x
)
differentiable at
x
=
1
x=1
x
=
1
? If so, find
f
′
(
1
)
f'(1)
f
′
(
1
)
.
f
(
x
)
=
{
x
+
1
if
x
<
1
1
2
x
2
+
3
2
if
x
≥
1
f(x) = \begin{cases} x+1 & \text{if } x < 1 \\ \frac{1}{2}x^2 + \frac{3}{2} & \text{if } x \geq 1 \end{cases}
f
(
x
)
=
{
x
+
1
2
1
x
2
+
2
3
if
x
<
1
if
x
≥
1
Differential Laws: nth Derivatives
If
y
=
(
10
e
+
1
)
10
y=\left(10e+1\right)^{10}
y
=
(
10
e
+
1
)
10
, find the 9
th
derivative of
y
y
y
.
If
f
(
x
)
f(x)
f
(
x
)
is differentiable everywhere, find
A
A
A
and
B
B
B
.
f
(
x
)
=
{
x
2
+
1
if
x
≥
0
A
x
+
B
if
x
<
0
f(x)=\left\{ \begin{array}{ll} \displaystyle x^2+1\quad\quad\,\,\,\,\,\text{if}\,x\geq 0 \\ Ax+B\quad\,\,\,\,\,\,\,\text{if}\,\, x<0\\ \end{array} \right.
f
(
x
)
=
{
x
2
+
1
if
x
≥
0
A
x
+
B
if
x
<
0