Question 1

Find the derivative of $f ( x ) = \ln ( x \sin 3 x )$

Accepted Answer

A)  $\frac { 1 } { 3 x } + 3 \tan 3 x$ 
B)  $\frac { 1 } { 3 x } + 3 \cot 3 x$ 
C)  $\frac { 1 } { x } + 3 \tan 3 x$ 
D)  $\frac { 1 } { x } + 3 \cot 3 x$ 
A)  $\frac { 1 } { 3 x } + 3 \tan 3 x$ 
B)  $\frac { 1 } { 3 x } + 3 \cot 3 x$ 
C)  $\frac { 1 } { x } + 3 \tan 3 x$ 
D)  $\frac { 1 } { x } + 3 \cot 3 x$ D

Question 2

Find the derivative of $f ( x ) = ( \sqrt [ 3 ] { x } + 2 \sqrt { x } ) ^ { 3 }$

Accepted Answer

$( \sqrt [ 3 ] { x } + 2 \sqrt { x } ) ^ { 2 } \left( \frac { 1 } { \sqrt [ 3 ] { x ^ { 2 } } } + \frac { 3 } { \sqrt { x } } \right)$

Question 3

Given $f ( x ) = \left\{ \begin{array} { l } 
- 2 x \text { if } x < 1 \
x - 3 \text { if } x \geq 1
\end{array} \right.$ , is $f$ continuous and/or differentiable at $x = 1 ?$ Explain.

Accepted Answer

A)  f is not continuous but differentiable at x = 1 
B)  f is neither continuous nor differentiable at x = 1 
C)  f is continuous but not differentiable at x = 1 
D)  f is both continuous and differentiable at x = 1 
A)  f is not continuous but differentiable at x = 1 
B)  f is neither continuous nor differentiable at x = 1 
C)  f is continuous but not differentiable at x = 1 
D)  f is both continuous and differentiable at x = 1 C

Question 4

Find the derivative of $f ( x ) = 5 - 2 e + 3 x - 2 x ^ { 5 } + \frac { 1 } { \sqrt [ 3 ] { x } }$

Accepted Answer

A)  $- 1 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 1 } { 3 } }$ 
B)  $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 1 } { 3 } }$ 
C)  $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 4 } { 3 } }$ 
D)  $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 2 } { 3 } }$ 
A)  $- 1 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 1 } { 3 } }$ 
B)  $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 1 } { 3 } }$ 
C)  $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 4 } { 3 } }$ 
D)  $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 2 } { 3 } }$

Question 5

Find the derivative of $f ( x ) = 2 x \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }$

Accepted Answer

A)  $\left( 112 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$ 
B)  $\left( 120 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }$ 
C)  $\left( 120 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$ 
D)  $112 x ^ { 2 } \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$ 
A)  $\left( 112 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$ 
B)  $\left( 120 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }$ 
C)  $\left( 120 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$ 
D)  $112 x ^ { 2 } \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$

Question 6

Differentiate $f ( x ) = \left( \frac { x ^ { 3 } - 1 } { \sqrt { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Accepted Answer

The answer of Differentiate \(f ( x ) = \left(...

Question 7

Find the derivative of $f ( x ) = e ^ { 5 x } \ln \left( x ^ { 2 } + 1 \right)$

Accepted Answer

The answer of Find the derivative of \(f ( x...

Question 8

Find the derivative of $f ( x ) = \frac { \sqrt [ 3 ] { x ^ { 5 } } - 3 x ^ { 5 } } { x ^ { 3 } }$

Accepted Answer

A)  $\frac { 4 } { 3 } x ^ { - \frac { 3 } { 5 } } - 6 x$ 
B)  $\frac { 4 } { 3 } x ^ { - \frac { 3 } { 7 } } - 6 x$ 
C)  $- \frac { 4 } { 3 } x ^ { - \frac { 7 } { 3 } } - 6 x$ 
D)  $\frac { 4 } { 3 } x ^ { - \frac { 7 } { 3 } } - 6 x$ 
A)  $\frac { 4 } { 3 } x ^ { - \frac { 3 } { 5 } } - 6 x$ 
B)  $\frac { 4 } { 3 } x ^ { - \frac { 3 } { 7 } } - 6 x$ 
C)  $- \frac { 4 } { 3 } x ^ { - \frac { 7 } { 3 } } - 6 x$ 
D)  $\frac { 4 } { 3 } x ^ { - \frac { 7 } { 3 } } - 6 x$

Question 9

Suppose that $h ( t )$ represents the height, in feet, of a person $t$ years old. In real world terms, what does $h ( 10 )$ represent? What is its unit? What does $h ^ { \prime } ( t )$ represents and what is its unit?

Accepted Answer

@#LAT-DLM& represents the height of a 10

Question 10

Find the derivative of $f ( x ) = \frac { \sqrt { 2 x + 1 } \left( x ^ { 2 } - 1 \right) ^ { 4 } } { ( x - 3 ) ^ { 3 } ( 2 + x ) }$

Accepted Answer

The answer of Find the derivative of \(f ( x...

Question 11

Find the derivative of $f ( x ) = \ln \left( 3 e ^ { 4 x } \right) + \sin ^ { 2 } x ^ { 3 }$

Accepted Answer

The answer of Find the derivative of \(f ( x...

Question 12

The function $f ( x ) = 9 - 2 x + x ^ { 2 }$ is both continuous and differentiable at $x = 0$ Write these facts as limit statements.

Accepted Answer

Since f is continuous at x = 0

Question 13

Find the derivative of $f ( x ) = \ln \left( \frac { x ^ { 5 } } { x ^ { 3 } + 2 x + 1 } \right)$

Accepted Answer

A)  $\frac { 5 } { x } + \frac { 3 x + 2 } { x ^ { 3 } + 2 x + 1 }$ 
B)  $\frac { 5 } { x } + \frac { 3 x ^ { 2 } + 2 } { x ^ { 3 } + 2 x + 1 }$ 
C)  $\frac { 5 } { x } - \frac { 3 x ^ { 2 } + 2 } { x ^ { 3 } + 2 x + 1 }$ 
D)  $\frac { 5 } { x } - \frac { 3 x ^ { 2 } - 2 } { x ^ { 3 } + 2 x + 1 }$ 
A)  $\frac { 5 } { x } + \frac { 3 x + 2 } { x ^ { 3 } + 2 x + 1 }$ 
B)  $\frac { 5 } { x } + \frac { 3 x ^ { 2 } + 2 } { x ^ { 3 } + 2 x + 1 }$ 
C)  $\frac { 5 } { x } - \frac { 3 x ^ { 2 } + 2 } { x ^ { 3 } + 2 x + 1 }$ 
D)  $\frac { 5 } { x } - \frac { 3 x ^ { 2 } - 2 } { x ^ { 3 } + 2 x + 1 }$

Question 14

Find the derivative of $f ( x ) = \sin ^ { 2 } x + 3 \cos ^ { 2 } x$

Accepted Answer

A)  $2 \sin x + 6 \cos x$ 
B)  $2 \sin 2 x$ 
C)  $4 \sin 2 x$ 
D)  $- 2 \sin 2 x$ 
A)  $2 \sin x + 6 \cos x$ 
B)  $2 \sin 2 x$ 
C)  $4 \sin 2 x$ 
D)  $- 2 \sin 2 x$

Question 15

Find the derivative of $f ( x ) = 2 x \sqrt { \sin 2 x \cos 2 x }$

Accepted Answer

The answer of Find the derivative of \(f ( x...

Question 16

Find the derivative of $f ( x ) = | 1 - 3 x |$

Accepted Answer

None

Question 17

Find the derivative of $f ( x ) = \cos ^ { - 1 } ( \ln x ) .$

Accepted Answer

A)  $\frac { 1 } { x \sqrt { 1 - \ln ^ { 2 } x } }$ 
B)  $\frac { 1 } { \sqrt { 1 - \ln ^ { 2 } x } }$ 
C)  $\frac { - 1 } { x \sqrt { 1 - \ln ^ { 2 } x } }$ 
D)  $\frac { - 1 } { \sqrt { 1 - \ln ^ { 2 } x } }$ 
A)  $\frac { 1 } { x \sqrt { 1 - \ln ^ { 2 } x } }$ 
B)  $\frac { 1 } { \sqrt { 1 - \ln ^ { 2 } x } }$ 
C)  $\frac { - 1 } { x \sqrt { 1 - \ln ^ { 2 } x } }$ 
D)  $\frac { - 1 } { \sqrt { 1 - \ln ^ { 2 } x } }$

Question 18

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( x ) \text {, if } f ( x ) = \frac { 1 } { x ^ { 2 } }$

Accepted Answer

The answer of Use the definition of derivative: \(\lim _...

Question 19

Find the derivative of $f ( x ) = 2 ^ { x } \tan x + 2 x$

Accepted Answer

A)  $x 2 ^ { x - 1 } \tan x + 2 ^ { x } \sec ^ { 2 } x + 2$ 
B)  $x 2 ^ { x - 1 } \tan x + 2 ^ { x } \sec x \tan x + 2$ 
C)  $2 ^ { x } \ln 2 \tan x + 2 ^ { x } \sec ^ { 2 } x + 2$ 
D)  $2 ^ { x } \ln 2 \tan x + 2 ^ { x } \sec x \tan x + 2$ 
A)  $x 2 ^ { x - 1 } \tan x + 2 ^ { x } \sec ^ { 2 } x + 2$ 
B)  $x 2 ^ { x - 1 } \tan x + 2 ^ { x } \sec x \tan x + 2$ 
C)  $2 ^ { x } \ln 2 \tan x + 2 ^ { x } \sec ^ { 2 } x + 2$ 
D)  $2 ^ { x } \ln 2 \tan x + 2 ^ { x } \sec x \tan x + 2$

Question 20

Differentiate $f ( x ) = \left( \frac { 2 x ^ { 3 } + 1 } { \sqrt [ 3 ] { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Accepted Answer

The answer of Differentiate \(f ( x ) = \left(...

Find the derivative of $f ( x ) = \ln ( x \sin 3 x )$

Find the derivative of $f ( x ) = ( \sqrt [ 3 ] { x } + 2 \sqrt { x } ) ^ { 3 }$

Given $f ( x ) = \left\{ \begin{array} { l } - 2 x \text { if } x < 1 \\x - 3 \text { if } x \geq 1\end{array} \right.$ , is $f$ continuous and/or differentiable at $x = 1 ?$ Explain.

Find the derivative of $f ( x ) = 5 - 2 e + 3 x - 2 x ^ { 5 } + \frac { 1 } { \sqrt [ 3 ] { x } }$

Find the derivative of $f ( x ) = 2 x \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }$

Differentiate $f ( x ) = \left( \frac { x ^ { 3 } - 1 } { \sqrt { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Find the derivative of $f ( x ) = e ^ { 5 x } \ln \left( x ^ { 2 } + 1 \right)$

Find the derivative of $f ( x ) = \frac { \sqrt [ 3 ] { x ^ { 5 } } - 3 x ^ { 5 } } { x ^ { 3 } }$

Suppose that $h ( t )$ represents the height, in feet, of a person $t$ years old. In real world terms, what does $h ( 10 )$ represent? What is its unit? What does $h ^ { \prime } ( t )$ represents and what is its unit?

Find the derivative of $f ( x ) = \frac { \sqrt { 2 x + 1 } \left( x ^ { 2 } - 1 \right) ^ { 4 } } { ( x - 3 ) ^ { 3 } ( 2 + x ) }$

Find the derivative of $f ( x ) = \ln \left( 3 e ^ { 4 x } \right) + \sin ^ { 2 } x ^ { 3 }$

The function $f ( x ) = 9 - 2 x + x ^ { 2 }$ is both continuous and differentiable at $x = 0$ Write these facts as limit statements.

Find the derivative of $f ( x ) = \ln \left( \frac { x ^ { 5 } } { x ^ { 3 } + 2 x + 1 } \right)$

Find the derivative of $f ( x ) = \sin ^ { 2 } x + 3 \cos ^ { 2 } x$

Find the derivative of $f ( x ) = 2 x \sqrt { \sin 2 x \cos 2 x }$

Find the derivative of $f ( x ) = | 1 - 3 x |$

Find the derivative of $f ( x ) = \cos ^ { - 1 } ( \ln x ) .$

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( x ) \text {, if } f ( x ) = \frac { 1 } { x ^ { 2 } }$

Find the derivative of $f ( x ) = 2 ^ { x } \tan x + 2 x$

Differentiate $f ( x ) = \left( \frac { 2 x ^ { 3 } + 1 } { \sqrt [ 3 ] { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Limits

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 2: Derivatives

Find the derivative of f(x)=ln⁡(xsin⁡3x)f ( x ) = \ln ( x \sin 3 x )f(x)=ln(xsin3x)

Find the derivative of f(x)=(x3+2x)3f ( x ) = ( \sqrt [ 3 ] { x } + 2 \sqrt { x } ) ^ { 3 }f(x)=(3x​+2x​)3

Given f(x)={−2x if x<1x−3 if x≥1f ( x ) = \left\{ \begin{array} { l } - 2 x \text { if } x < 1 \\x - 3 \text { if } x \geq 1\end{array} \right.f(x)={−2x if x<1x−3 if x≥1​ , is fff continuous and/or differentiable at x=1?x = 1 ?x=1? Explain.

Find the derivative of f(x)=5−2e+3x−2x5+1x3f ( x ) = 5 - 2 e + 3 x - 2 x ^ { 5 } + \frac { 1 } { \sqrt [ 3 ] { x } }f(x)=5−2e+3x−2x5+3x​1​

Find the derivative of f(x)=2x(4x2+1)7f ( x ) = 2 x \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }f(x)=2x(4x2+1)7

Differentiate f(x)=(x3−1x)3f ( x ) = \left( \frac { x ^ { 3 } - 1 } { \sqrt { x } } \right) ^ { 3 }f(x)=(x​x3−1​)3 in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Find the derivative of f(x)=e5xln⁡(x2+1)f ( x ) = e ^ { 5 x } \ln \left( x ^ { 2 } + 1 \right)f(x)=e5xln(x2+1)

Find the derivative of f(x)=x53−3x5x3f ( x ) = \frac { \sqrt [ 3 ] { x ^ { 5 } } - 3 x ^ { 5 } } { x ^ { 3 } }f(x)=x33x5​−3x5​

Suppose that h(t)h ( t )h(t) represents the height, in feet, of a person ttt years old. In real world terms, what does h(10)h ( 10 )h(10) represent? What is its unit? What does h′(t)h ^ { \prime } ( t )h′(t) represents and what is its unit?

Find the derivative of f(x)=2x+1(x2−1)4(x−3)3(2+x)f ( x ) = \frac { \sqrt { 2 x + 1 } \left( x ^ { 2 } - 1 \right) ^ { 4 } } { ( x - 3 ) ^ { 3 } ( 2 + x ) }f(x)=(x−3)3(2+x)2x+1​(x2−1)4​

Find the derivative of f(x)=ln⁡(3e4x)+sin⁡2x3f ( x ) = \ln \left( 3 e ^ { 4 x } \right) + \sin ^ { 2 } x ^ { 3 }f(x)=ln(3e4x)+sin2x3

The function f(x)=9−2x+x2f ( x ) = 9 - 2 x + x ^ { 2 }f(x)=9−2x+x2 is both continuous and differentiable at x=0x = 0x=0 Write these facts as limit statements.

Find the derivative of f(x)=ln⁡(x5x3+2x+1)f ( x ) = \ln \left( \frac { x ^ { 5 } } { x ^ { 3 } + 2 x + 1 } \right)f(x)=ln(x3+2x+1x5​)

Find the derivative of f(x)=sin⁡2x+3cos⁡2xf ( x ) = \sin ^ { 2 } x + 3 \cos ^ { 2 } xf(x)=sin2x+3cos2x

Find the derivative of f(x)=2xsin⁡2xcos⁡2xf ( x ) = 2 x \sqrt { \sin 2 x \cos 2 x }f(x)=2xsin2xcos2x​

Find the derivative of f(x)=∣1−3x∣f ( x ) = | 1 - 3 x |f(x)=∣1−3x∣

Find the derivative of f(x)=cos⁡−1(ln⁡x).f ( x ) = \cos ^ { - 1 } ( \ln x ) .f(x)=cos−1(lnx).

Use the definition of derivative: lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ to find f′(x), if f(x)=1x2f ^ { \prime } ( x ) \text {, if } f ( x ) = \frac { 1 } { x ^ { 2 } }f′(x), if f(x)=x21​

Find the derivative of f(x)=2xtan⁡x+2xf ( x ) = 2 ^ { x } \tan x + 2 xf(x)=2xtanx+2x

Differentiate f(x)=(2x3+1x3)3f ( x ) = \left( \frac { 2 x ^ { 3 } + 1 } { \sqrt [ 3 ] { x } } \right) ^ { 3 }f(x)=(3x​2x3+1​)3 in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Limits

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Find the derivative of $f ( x ) = \ln ( x \sin 3 x )$

Find the derivative of $f ( x ) = ( \sqrt [ 3 ] { x } + 2 \sqrt { x } ) ^ { 3 }$

Given $f ( x ) = \left\{ \begin{array} { l } - 2 x \text { if } x < 1 \\x - 3 \text { if } x \geq 1\end{array} \right.$ , is $f$ continuous and/or differentiable at $x = 1 ?$ Explain.

Find the derivative of $f ( x ) = 5 - 2 e + 3 x - 2 x ^ { 5 } + \frac { 1 } { \sqrt [ 3 ] { x } }$

Find the derivative of $f ( x ) = 2 x \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }$

Differentiate $f ( x ) = \left( \frac { x ^ { 3 } - 1 } { \sqrt { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Find the derivative of $f ( x ) = e ^ { 5 x } \ln \left( x ^ { 2 } + 1 \right)$

Find the derivative of $f ( x ) = \frac { \sqrt [ 3 ] { x ^ { 5 } } - 3 x ^ { 5 } } { x ^ { 3 } }$

Suppose that $h ( t )$ represents the height, in feet, of a person $t$ years old. In real world terms, what does $h ( 10 )$ represent? What is its unit? What does $h ^ { \prime } ( t )$ represents and what is its unit?

Find the derivative of $f ( x ) = \frac { \sqrt { 2 x + 1 } \left( x ^ { 2 } - 1 \right) ^ { 4 } } { ( x - 3 ) ^ { 3 } ( 2 + x ) }$

Find the derivative of $f ( x ) = \ln \left( 3 e ^ { 4 x } \right) + \sin ^ { 2 } x ^ { 3 }$

The function $f ( x ) = 9 - 2 x + x ^ { 2 }$ is both continuous and differentiable at $x = 0$ Write these facts as limit statements.

Find the derivative of $f ( x ) = \ln \left( \frac { x ^ { 5 } } { x ^ { 3 } + 2 x + 1 } \right)$

Find the derivative of $f ( x ) = \sin ^ { 2 } x + 3 \cos ^ { 2 } x$

Find the derivative of $f ( x ) = 2 x \sqrt { \sin 2 x \cos 2 x }$

Find the derivative of $f ( x ) = | 1 - 3 x |$

Find the derivative of $f ( x ) = \cos ^ { - 1 } ( \ln x ) .$

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( x ) \text {, if } f ( x ) = \frac { 1 } { x ^ { 2 } }$

Find the derivative of $f ( x ) = 2 ^ { x } \tan x + 2 x$

Differentiate $f ( x ) = \left( \frac { 2 x ^ { 3 } + 1 } { \sqrt [ 3 ] { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.