Question 1

Let $f ( x ) = \cosh \left( 3 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\ln 3 \sinh \left( 3 ^ { x } \right)$ 
B)  $3 ^ { x } \sinh \left( 3 ^ { x } \right)$ 
C)  $\sinh \left( 3 ^ { x } \right)$ 
D)  $3 ^ { x } \ln 3 \sinh \left( 3 ^ { x } \right)$ 
E)  $- 3 ^ { x } \ln 3 \sinh \left( 3 ^ { x } \right)$ 
A)  $\ln 3 \sinh \left( 3 ^ { x } \right)$ 
B)  $3 ^ { x } \sinh \left( 3 ^ { x } \right)$ 
C)  $\sinh \left( 3 ^ { x } \right)$ 
D)  $3 ^ { x } \ln 3 \sinh \left( 3 ^ { x } \right)$ 
E)  $- 3 ^ { x } \ln 3 \sinh \left( 3 ^ { x } \right)$

Question 2

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25$ the approximation of $\sqrt { 24 }$ by differentials is

Accepted Answer

A) 5.1 
B) 5.05 
C) 4.95 
D) 4.9 
E) 4.85 
A) 5.1 
B) 5.05 
C) 4.95 
D) 4.9 
E) 4.85

Question 3

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25 ,$ the approximation of $\sqrt { 25.1 }$ by differentials is

Accepted Answer

A) 5.1 
B) 5.05 
C) 5.01 
D) 4.99 
E) 4.95 
A) 5.1 
B) 5.05 
C) 5.01 
D) 4.99 
E) 4.95

Question 4

Let $x ^ { y } = e \text {. }$ Then $y ^ { \prime }$ is

Accepted Answer

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

Question 5

Let $f ( x ) = \tan ^ { 3 } ( 5 x )$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $15 \tan ^ { 2 } ( 5 x )$ 
B)  $15 \tan ^ { 2 } ( 5 x ) \sec ( 5 x )$ 
C)  $15 \tan ^ { 2 } ( 5 x ) \sec ^ { 2 } ( 5 x )$ 
D)  $5 \tan ^ { 3 } ( 5 x ) \sec ^ { 2 } ( 5 x )$ 
E)  $5 \tan ^ { 2 } ( 5 x ) \sec ( 5 x )$ 
A)  $15 \tan ^ { 2 } ( 5 x )$ 
B)  $15 \tan ^ { 2 } ( 5 x ) \sec ( 5 x )$ 
C)  $15 \tan ^ { 2 } ( 5 x ) \sec ^ { 2 } ( 5 x )$ 
D)  $5 \tan ^ { 3 } ( 5 x ) \sec ^ { 2 } ( 5 x )$ 
E)  $5 \tan ^ { 2 } ( 5 x ) \sec ( 5 x )$

Question 6

Let $y = \tan ^ { 3 } x$ Then $y ^ { \prime }$ is

Accepted Answer

A)  $3 \tan ^ { 2 } x$ 
B)  $3 \tan x \sec ^ { 2 } x$ 
C)  $3 \tan ^ { 2 } x \sec ^ { 2 } x$ 
D)  $3 \sec ^ { 2 } x$ 
E)  $3 \tan ^ { 2 } x \sec ^ { 4 } x$ 
A)  $3 \tan ^ { 2 } x$ 
B)  $3 \tan x \sec ^ { 2 } x$ 
C)  $3 \tan ^ { 2 } x \sec ^ { 2 } x$ 
D)  $3 \sec ^ { 2 } x$ 
E)  $3 \tan ^ { 2 } x \sec ^ { 4 } x$

Question 7

Let $f ( x ) = \sinh ( \sin x )$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\sinh ( \cos x )$ 
B)  $\cos x [ \cosh ( \sin x ) ]$ 
C)  $\sin x [ \cosh ( \cos x ) ]$ 
D)  $\cos x [ \sinh ( \sin x ) ]$ 
E)  $\sin x [ \sinh ( \sin x ) ]$ 
A)  $\sinh ( \cos x )$ 
B)  $\cos x [ \cosh ( \sin x ) ]$ 
C)  $\sin x [ \cosh ( \cos x ) ]$ 
D)  $\cos x [ \sinh ( \sin x ) ]$ 
E)  $\sin x [ \sinh ( \sin x ) ]$

Question 8

Let $f ( x ) = \tan ^ { - 1 } ( \sqrt { x } )$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\frac { 1 } { 2 \sqrt { x } \left( 1 + x ^ { 2 } \right) }$ 
B)  $\frac { 2 } { \sqrt { x } \left( 1 + x ^ { 2 } \right) }$ 
C)  $\frac { 1 } { 2 \sqrt { x } \left( 1 - x ^ { 2 } \right) }$ 
D)  $\frac { 1 } { 2 \sqrt { x } ( 1 + x ) }$ 
E)  $\frac { 1 } { 2 \sqrt { x } ( 1 - x ) }$ 
A)  $\frac { 1 } { 2 \sqrt { x } \left( 1 + x ^ { 2 } \right) }$ 
B)  $\frac { 2 } { \sqrt { x } \left( 1 + x ^ { 2 } \right) }$ 
C)  $\frac { 1 } { 2 \sqrt { x } \left( 1 - x ^ { 2 } \right) }$ 
D)  $\frac { 1 } { 2 \sqrt { x } ( 1 + x ) }$ 
E)  $\frac { 1 } { 2 \sqrt { x } ( 1 - x ) }$

Question 9

Let $f ( x ) = \cosh \left( x ^ { 2 } + 1 \right)$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $- 2 x \sinh \left( x ^ { 2 } + 1 \right)$ 
B)  $2 x \sinh \left( x ^ { 2 } + 1 \right)$ 
C)  $\sinh ( 2 x )$ 
D)  $- \sinh ( 2 x )$ 
E)  $x ^ { 2 } \sinh \left( x ^ { 2 } + 1 \right)$ 
A)  $- 2 x \sinh \left( x ^ { 2 } + 1 \right)$ 
B)  $2 x \sinh \left( x ^ { 2 } + 1 \right)$ 
C)  $\sinh ( 2 x )$ 
D)  $- \sinh ( 2 x )$ 
E)  $x ^ { 2 } \sinh \left( x ^ { 2 } + 1 \right)$

Question 10

Let $f ( x ) = \sinh ^ { - 1 } ( \sin x )$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\frac { \cos x } { \sqrt { \sin ^ { 2 } x + 1 } }$ 
B)  $\frac { \cos x } { \sqrt { \sin ^ { 2 } x - 1 } }$ 
C)  $\frac { \cos x } { \sqrt { 1 - \sin ^ { 2 } x } }$ 
D)  $\frac { - \cos x } { \sqrt { \sin ^ { 2 } x + 1 } }$ 
E)  $\frac { - \cos x } { \sqrt { \sin ^ { 2 } x - 1 } }$ 
A)  $\frac { \cos x } { \sqrt { \sin ^ { 2 } x + 1 } }$ 
B)  $\frac { \cos x } { \sqrt { \sin ^ { 2 } x - 1 } }$ 
C)  $\frac { \cos x } { \sqrt { 1 - \sin ^ { 2 } x } }$ 
D)  $\frac { - \cos x } { \sqrt { \sin ^ { 2 } x + 1 } }$ 
E)  $\frac { - \cos x } { \sqrt { \sin ^ { 2 } x - 1 } }$

Question 11

Let $f ( x ) = 2 ^ { x } \cosh x$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $2 ^ { x } [ \sinh x + \cosh x ] 2 ^ { x } [ \sinh x + \cosh x ]$ 
B)  $2 ^ { x } [ \sinh x + \ln 2 \cosh x ]$ 
C)  $2 ^ { x } [ \ln 2 \sinh x + \cosh x ]$ 
D)  $2 ^ { x } [ \sinh x - \ln 2 \cosh x ]$ 
E)  $2 ^ { x } [ \ln 2 \sinh x - \cosh x ]$ 
A)  $2 ^ { x } [ \sinh x + \cosh x ] 2 ^ { x } [ \sinh x + \cosh x ]$ 
B)  $2 ^ { x } [ \sinh x + \ln 2 \cosh x ]$ 
C)  $2 ^ { x } [ \ln 2 \sinh x + \cosh x ]$ 
D)  $2 ^ { x } [ \sinh x - \ln 2 \cosh x ]$ 
E)  $2 ^ { x } [ \ln 2 \sinh x - \cosh x ]$

Question 12

Let $y = \log _ { 2 } \left( \tan ^ { 3 } x \right)$ Then dy is

Accepted Answer

A)  $\frac { 3 \sec x \sin x } { \ln 2 } d x$ 
B)  $\frac { 3 \cos x \csc x } { \ln 2 } d x$ 
C)  $\frac { 3 \sec x \csc x } { \ln 2 } d x$ 
D)  $\frac { 3 \cos x \sin x } { \ln 2 } d x$ 
E)  $\frac { \cos x \sin x } { 3 \ln 2 } d x$ 
A)  $\frac { 3 \sec x \sin x } { \ln 2 } d x$ 
B)  $\frac { 3 \cos x \csc x } { \ln 2 } d x$ 
C)  $\frac { 3 \sec x \csc x } { \ln 2 } d x$ 
D)  $\frac { 3 \cos x \sin x } { \ln 2 } d x$ 
E)  $\frac { \cos x \sin x } { 3 \ln 2 } d x$

Question 13

Let $f ( x ) = x ^ { 2 } \sin ^ { - 1 } x$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

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

Question 14

Let $h = f \circ g .$ Then $g ( 2 ) = 3$ , $g ^ { \prime } ( 2 ) = - 1$ and $f ^ { \prime } ( 3 ) = 6$  Then $h ^ { \prime } ( 2 )$ is

Accepted Answer

A) -6 
B) 6 
C) -3 
D) 18 
E) 16 
A) -6 
B) 6 
C) -3 
D) 18 
E) 16

Question 15

Using implicit differentiation on $y = ( x + 2 y ) ^ { 2 } , y ^ { \prime }$ is

Accepted Answer

A)  $\frac { x + 2 y } { 1 - 2 ( x + 2 y ) }$ 
B)  $\frac { 2 ( x + 2 y ) } { 4 ( x + 2 y ) - 1 }$ 
C)  $\frac { 2 ( x + 2 y ) } { 1 - 4 ( x + 2 y ) }$ 
D)  $\frac { 2 ( x + 2 y ) } { 1 + 4 ( x + 2 y ) }$ 
E)  $\frac { 2 ( x - 2 y ) } { 1 - 4 ( x + 2 y ) }$ 
A)  $\frac { x + 2 y } { 1 - 2 ( x + 2 y ) }$ 
B)  $\frac { 2 ( x + 2 y ) } { 4 ( x + 2 y ) - 1 }$ 
C)  $\frac { 2 ( x + 2 y ) } { 1 - 4 ( x + 2 y ) }$ 
D)  $\frac { 2 ( x + 2 y ) } { 1 + 4 ( x + 2 y ) }$ 
E)  $\frac { 2 ( x - 2 y ) } { 1 - 4 ( x + 2 y ) }$

Question 16

Let $y = \ln ( x y )$ Using implicit differentiation, $y ^ { \prime }$ is

Accepted Answer

A)  $\frac { x } { y ( y - 1 ) }$ 
B)  $\frac { x } { y ( 1 - y ) }$ 
C)  $\frac { y } { x ( y - 1 ) }$ 
D)  $\frac { y } { x ( 1 - y ) }$ 
E)  $\frac { x } { y ( x - 1 ) }$ 
A)  $\frac { x } { y ( y - 1 ) }$ 
B)  $\frac { x } { y ( 1 - y ) }$ 
C)  $\frac { y } { x ( y - 1 ) }$ 
D)  $\frac { y } { x ( 1 - y ) }$ 
E)  $\frac { x } { y ( x - 1 ) }$

Question 17

Let $f ( x ) = \operatorname { coth } ( 3 x )$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $3 \operatorname { csch } ^ { 2 } ( 3 x )$ 
B)  $- 3 \operatorname { csch } ^ { 2 } ( x )$ 
C)  $3 \operatorname { csch } ^ { 2 } ( x )$ 
D)  $- 3 \operatorname { coth } ^ { 2 } ( 3 x )$ 
E)  $- 3 \operatorname { csch } ^ { 2 } ( 3 x )$ 
A)  $3 \operatorname { csch } ^ { 2 } ( 3 x )$ 
B)  $- 3 \operatorname { csch } ^ { 2 } ( x )$ 
C)  $3 \operatorname { csch } ^ { 2 } ( x )$ 
D)  $- 3 \operatorname { coth } ^ { 2 } ( 3 x )$ 
E)  $- 3 \operatorname { csch } ^ { 2 } ( 3 x )$

Question 18

Let $y = \sin u + \cos u$ and $u = 4 x ^ { 2 } + 5$ By the Chain Rule, $\frac { d y } { d x }$ is

Accepted Answer

A)  $\cos ( 8 x ) - \sin ( 8 x )$ 
B)  $\cos ( 8 x ) + \sin ( 8 x )$ 
C)  $\cos \left( 4 x ^ { 2 } + 5 \right) - \sin \left( 4 x ^ { 2 } + 5 \right)$ 
D)  $\cos \left( 4 x ^ { 2 } + 5 \right) + \sin \left( 4 x ^ { 2 } + 5 \right)$ 
E)  $8 x \left[ \cos \left( 4 x ^ { 2 } + 5 \right) - \sin \left( 4 x ^ { 2 } + 5 \right) \right]$ 
A)  $\cos ( 8 x ) - \sin ( 8 x )$ 
B)  $\cos ( 8 x ) + \sin ( 8 x )$ 
C)  $\cos \left( 4 x ^ { 2 } + 5 \right) - \sin \left( 4 x ^ { 2 } + 5 \right)$ 
D)  $\cos \left( 4 x ^ { 2 } + 5 \right) + \sin \left( 4 x ^ { 2 } + 5 \right)$ 
E)  $8 x \left[ \cos \left( 4 x ^ { 2 } + 5 \right) - \sin \left( 4 x ^ { 2 } + 5 \right) \right]$

Question 19

Let $f ( x ) = \ln \left( \log _ { 2 } x \right)$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\frac { 1 } { x \log _ { 2 } x }$ 
B)  $\frac { \ln 2 } { x \log _ { 2 } x }$ 
C)  $\frac { 1 } { x \ln x }$ 
D)  $- \frac { 1 } { x \ln 2 ( \ln x ) }$ 
E)  $\frac { \ln 2 } { x \ln x }$ 
A)  $\frac { 1 } { x \log _ { 2 } x }$ 
B)  $\frac { \ln 2 } { x \log _ { 2 } x }$ 
C)  $\frac { 1 } { x \ln x }$ 
D)  $- \frac { 1 } { x \ln 2 ( \ln x ) }$ 
E)  $\frac { \ln 2 } { x \ln x }$

Question 20

Let $y = \ln \left( 5 ^ { x } \right)$ Then dy is

Accepted Answer

A)  $\frac { 1 } { \ln 5 } d x$ 
B)  $\frac { 1 } { 5 ^ { x } } d x$ 
C)  $5 ^ { x } \ln 5 d x$ 
D)  $\frac { 5 ^ { x } } { \ln 5 } d x$ 
E)  $\ln 5 d x$ 
A)  $\frac { 1 } { \ln 5 } d x$ 
B)  $\frac { 1 } { 5 ^ { x } } d x$ 
C)  $5 ^ { x } \ln 5 d x$ 
D)  $\frac { 5 ^ { x } } { \ln 5 } d x$ 
E)  $\ln 5 d x$

Let $f ( x ) = \cosh \left( 3 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25$ the approximation of $\sqrt { 24 }$ by differentials is

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25 ,$ the approximation of $\sqrt { 25.1 }$ by differentials is

Let $x ^ { y } = e \text {. }$ Then $y ^ { \prime }$ is

Let $f ( x ) = \tan ^ { 3 } ( 5 x )$ Then $f ^ { \prime } ( x )$ is

Let $y = \tan ^ { 3 } x$ Then $y ^ { \prime }$ is

Let $f ( x ) = \sinh ( \sin x )$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \tan ^ { - 1 } ( \sqrt { x } )$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \cosh \left( x ^ { 2 } + 1 \right)$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \sinh ^ { - 1 } ( \sin x )$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = 2 ^ { x } \cosh x$ Then $f ^ { \prime } ( x )$ is

Let $y = \log _ { 2 } \left( \tan ^ { 3 } x \right)$ Then dy is

Let $f ( x ) = x ^ { 2 } \sin ^ { - 1 } x$ Then $f ^ { \prime } ( x )$ is

Let $h = f \circ g .$ Then $g ( 2 ) = 3$ , $g ^ { \prime } ( 2 ) = - 1$ and $f ^ { \prime } ( 3 ) = 6$ Then $h ^ { \prime } ( 2 )$ is

Using implicit differentiation on $y = ( x + 2 y ) ^ { 2 } , y ^ { \prime }$ is

Let $y = \ln ( x y )$ Using implicit differentiation, $y ^ { \prime }$ is

Let $f ( x ) = \operatorname { coth } ( 3 x )$ Then $f ^ { \prime } ( x )$ is

Let $y = \sin u + \cos u$ and $u = 4 x ^ { 2 } + 5$ By the Chain Rule, $\frac { d y } { d x }$ is

Let $f ( x ) = \ln \left( \log _ { 2 } x \right)$ Then $f ^ { \prime } ( x )$ is

Let $y = \ln \left( 5 ^ { x } \right)$ Then dy is

Preparing for Calculus

Limits and Continuity

The Derivative

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

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Exam 4: More About Derivatives

Let f(x)=cosh⁡(3x)f ( x ) = \cosh \left( 3 ^ { x } \right)f(x)=cosh(3x) Then f′(x)f ^ { \prime } ( x )f′(x) is

Letting f(x)=xf ( x ) = \sqrt { x }f(x)=x​ and x0=25x _ { 0 } = 25x0​=25 the approximation of 24\sqrt { 24 }24​ by differentials is

Letting f(x)=xf ( x ) = \sqrt { x }f(x)=x​ and x0=25,x _ { 0 } = 25 ,x0​=25, the approximation of 25.1\sqrt { 25.1 }25.1​ by differentials is

Let xy=e. x ^ { y } = e \text {. }xy=e. Then y′y ^ { \prime }y′ is

Let f(x)=tan⁡3(5x)f ( x ) = \tan ^ { 3 } ( 5 x )f(x)=tan3(5x) Then f′(x)f ^ { \prime } ( x )f′(x) is

Let y=tan⁡3xy = \tan ^ { 3 } xy=tan3x Then y′y ^ { \prime }y′ is

Let f(x)=sinh⁡(sin⁡x)f ( x ) = \sinh ( \sin x )f(x)=sinh(sinx) Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=tan⁡−1(x)f ( x ) = \tan ^ { - 1 } ( \sqrt { x } )f(x)=tan−1(x​) Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=cosh⁡(x2+1)f ( x ) = \cosh \left( x ^ { 2 } + 1 \right)f(x)=cosh(x2+1) Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=sinh⁡−1(sin⁡x)f ( x ) = \sinh ^ { - 1 } ( \sin x )f(x)=sinh−1(sinx) Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=2xcosh⁡xf ( x ) = 2 ^ { x } \cosh xf(x)=2xcoshx Then f′(x)f ^ { \prime } ( x )f′(x) is

Let y=log⁡2(tan⁡3x)y = \log _ { 2 } \left( \tan ^ { 3 } x \right)y=log2​(tan3x) Then dy is

Let f(x)=x2sin⁡−1xf ( x ) = x ^ { 2 } \sin ^ { - 1 } xf(x)=x2sin−1x Then f′(x)f ^ { \prime } ( x )f′(x) is

Let h=f∘g.h = f \circ g .h=f∘g. Then g(2)=3g ( 2 ) = 3g(2)=3 , g′(2)=−1g ^ { \prime } ( 2 ) = - 1g′(2)=−1 and f′(3)=6f ^ { \prime } ( 3 ) = 6f′(3)=6 Then h′(2)h ^ { \prime } ( 2 )h′(2) is

Using implicit differentiation on y=(x+2y)2,y′y = ( x + 2 y ) ^ { 2 } , y ^ { \prime }y=(x+2y)2,y′ is

Let y=ln⁡(xy)y = \ln ( x y )y=ln(xy) Using implicit differentiation, y′y ^ { \prime }y′ is

Let f(x)=coth⁡(3x)f ( x ) = \operatorname { coth } ( 3 x )f(x)=coth(3x) Then f′(x)f ^ { \prime } ( x )f′(x) is

Let y=sin⁡u+cos⁡uy = \sin u + \cos uy=sinu+cosu and u=4x2+5u = 4 x ^ { 2 } + 5u=4x2+5 By the Chain Rule, dydx\frac { d y } { d x }dxdy​ is

Let f(x)=ln⁡(log⁡2x)f ( x ) = \ln \left( \log _ { 2 } x \right)f(x)=ln(log2​x) Then f′(x)f ^ { \prime } ( x )f′(x) is

Let y=ln⁡(5x)y = \ln \left( 5 ^ { x } \right)y=ln(5x) Then dy is

Preparing for Calculus

Limits and Continuity

The Derivative

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Let $f ( x ) = \cosh \left( 3 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25$ the approximation of $\sqrt { 24 }$ by differentials is

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25 ,$ the approximation of $\sqrt { 25.1 }$ by differentials is

Let $x ^ { y } = e \text {. }$ Then $y ^ { \prime }$ is

Let $f ( x ) = \tan ^ { 3 } ( 5 x )$ Then $f ^ { \prime } ( x )$ is

Let $y = \tan ^ { 3 } x$ Then $y ^ { \prime }$ is

Let $f ( x ) = \sinh ( \sin x )$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \tan ^ { - 1 } ( \sqrt { x } )$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \cosh \left( x ^ { 2 } + 1 \right)$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \sinh ^ { - 1 } ( \sin x )$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = 2 ^ { x } \cosh x$ Then $f ^ { \prime } ( x )$ is

Let $y = \log _ { 2 } \left( \tan ^ { 3 } x \right)$ Then dy is

Let $f ( x ) = x ^ { 2 } \sin ^ { - 1 } x$ Then $f ^ { \prime } ( x )$ is

Let $h = f \circ g .$ Then $g ( 2 ) = 3$ , $g ^ { \prime } ( 2 ) = - 1$ and $f ^ { \prime } ( 3 ) = 6$ Then $h ^ { \prime } ( 2 )$ is

Using implicit differentiation on $y = ( x + 2 y ) ^ { 2 } , y ^ { \prime }$ is

Let $y = \ln ( x y )$ Using implicit differentiation, $y ^ { \prime }$ is

Let $f ( x ) = \operatorname { coth } ( 3 x )$ Then $f ^ { \prime } ( x )$ is

Let $y = \sin u + \cos u$ and $u = 4 x ^ { 2 } + 5$ By the Chain Rule, $\frac { d y } { d x }$ is

Let $f ( x ) = \ln \left( \log _ { 2 } x \right)$ Then $f ^ { \prime } ( x )$ is

Let $y = \ln \left( 5 ^ { x } \right)$ Then dy is