Question 1

If f(x) = x $( x - 1 ) ^ { 5 }$ and g(x) = $x ^ { 2 }$ , find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)).

Accepted Answer

A)  2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 2 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$ 
B)  $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 5 $x ^ { 2 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$ 
C)  2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 3 }$ ( $x ^ { 2 }$ - 1) 
D)  2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 3 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$ 
A)  2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 2 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$ 
B)  $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 5 $x ^ { 2 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$ 
C)  2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 3 }$ ( $x ^ { 2 }$ - 1) 
D)  2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 3 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$

Question 2

Find the equation of the tangent line to the graph of $x ^ { 2 }$ + $y ^ { 2 }$ = 1 at the point $\left( \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ .
Is $y = - \frac { 1 } { \sqrt { 3 } } x + \frac { 2 \sqrt { 3 } } { 3 }$ correct?

Accepted Answer

A) True 
 B)False

Question 3

Differentiate
-F(r) = $\mathrm { r } ^ { 2 }$ (r - 1)(r + 1 $) ^ { - 1 }$

Accepted Answer

A)  (2 $r ^ { 3 }$ + 2 $\mathrm { r } ^ { 2 }$ - 2r)(r + 1 $) ^ { - 2 }$ 
B)  2r( $\mathrm { r } ^ { 2 }$ + r -1) 
C)  $\frac { - 2 \mathrm { r } } { ( \mathrm { r } + 1 ) ^ { 2 } }$ 
D)  4 $r ^ { 3 }$ - 2r 
A)  (2 $r ^ { 3 }$ + 2 $\mathrm { r } ^ { 2 }$ - 2r)(r + 1 $) ^ { - 2 }$ 
B)  2r( $\mathrm { r } ^ { 2 }$ + r -1) 
C)  $\frac { - 2 \mathrm { r } } { ( \mathrm { r } + 1 ) ^ { 2 } }$ 
D)  4 $r ^ { 3 }$ - 2r

Question 4

Assume 4 $x ^ { 3 }$ + 2xy - $y ^ { 3 }$ = $\frac { 1 } { 2 }$ . What is the slope of the graph at the point $\left( \frac { 1 } { 2 } , - 1 \right)$ ?

Accepted Answer

A)  - $\frac { 1 } { 2 }$ 
B)  8 
C)  $\frac { 8 } { 5 }$ 
D)  1 
A)  - $\frac { 1 } { 2 }$ 
B)  8 
C)  $\frac { 8 } { 5 }$ 
D)  1

Question 5

Let g(x) = $\sqrt { x }$ . Using the chain rule, find an expression for the derivative of [g(f(x))].

Accepted Answer

A)  $\frac { 1 } { 2 \sqrt { f ( x ) } }$ f'(x) 
B)  $\frac { 1 } { 2 \sqrt { x } }$ f'( $\sqrt { x }$ ) 
C)  $\sqrt { x }$ f'(x) + $\frac { 1 } { 2 }$ $x ^ { - 1 / 2 }$ f(x) 
D)  $\sqrt { x }$ f( $\sqrt { x }$ ) 
E)  none of these 
A)  $\frac { 1 } { 2 \sqrt { f ( x ) } }$ f'(x) 
B)  $\frac { 1 } { 2 \sqrt { x } }$ f'( $\sqrt { x }$ ) 
C)  $\sqrt { x }$ f'(x) + $\frac { 1 } { 2 }$ $x ^ { - 1 / 2 }$ f(x) 
D)  $\sqrt { x }$ f( $\sqrt { x }$ ) 
E)  none of these

Question 6

Differentiate
-f(x) = ( $x ^ { 2 }$ + 1 $) ^ { 3 }$ ( $x ^ { 3 }$ - 1 $)^2$ at x = -1
Enter just an integer.

Accepted Answer

The answer of Differentiate
-f(x) = ( \(x ^ { 2...

Question 7

Use the chain rule to compute the derivative of $\frac { ( 3 x + 1 ) ^ { 3 } } { ( 3 x + 1 ) ^ { 3 } + 1 }$ at x = -1.
Enter just a reduced fraction.

Accepted Answer

The answer of Use the chain rule to compute the...

Question 8

If y = $\sqrt { u }$ and u = $x ^ { 3 }$ - 5 $x ^ { 2 }$ + 1, find $\frac { d y } { d x }$ .

Accepted Answer

A)  $\sqrt { 3 x ^ { 2 } - 10 x }$ 
B)  $\sqrt { 3 x ^ { 2 } - 10 x }$ $\left( x ^ { 3 } \right.$ - $5 x ^ { 2 }$ + 1) 
C)  $\frac { 3 x ^ { 2 } - 10 x } { 2 \sqrt { x ^ { 3 } - 5 x ^ { 2 } + 1 } }$ 
D)  $\sqrt { x ^ { 3 } - 5 x ^ { 2 } + 1 }$ 
A)  $\sqrt { 3 x ^ { 2 } - 10 x }$ 
B)  $\sqrt { 3 x ^ { 2 } - 10 x }$ $\left( x ^ { 3 } \right.$ - $5 x ^ { 2 }$ + 1) 
C)  $\frac { 3 x ^ { 2 } - 10 x } { 2 \sqrt { x ^ { 3 } - 5 x ^ { 2 } + 1 } }$ 
D)  $\sqrt { x ^ { 3 } - 5 x ^ { 2 } + 1 }$

Question 9

Compute $\frac { \mathrm { dy } } { \mathrm { dx } }$ y = 4 $u ^ { 2 }$ + 8u + 4 and u = 3x + 1.
Enter your answer as just a polynomial in x in standard form (no label).

Accepted Answer

The answer of Compute \(\frac { \mathrm { dy }...

If f(x) = x $( x - 1 ) ^ { 5 }$ and g(x) = $x ^ { 2 }$ , find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)).

Find the equation of the tangent line to the graph of $x ^ { 2 }$ + $y ^ { 2 }$ = 1 at the point $\left( \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ . Is $y = - \frac { 1 } { \sqrt { 3 } } x + \frac { 2 \sqrt { 3 } } { 3 }$ correct?

Differentiate -F(r) = $\mathrm { r } ^ { 2 }$ (r - 1)(r + 1 $) ^ { - 1 }$

Assume 4 $x ^ { 3 }$ + 2xy - $y ^ { 3 }$ = $\frac { 1 } { 2 }$ . What is the slope of the graph at the point $\left( \frac { 1 } { 2 } , - 1 \right)$ ?

Let g(x) = $\sqrt { x }$ . Using the chain rule, find an expression for the derivative of [g(f(x))].

Differentiate -f(x) = ( $x ^ { 2 }$ + 1 $) ^ { 3 }$ ( $x ^ { 3 }$ - 1 $)^2$ at x = -1 Enter just an integer.

Use the chain rule to compute the derivative of $\frac { ( 3 x + 1 ) ^ { 3 } } { ( 3 x + 1 ) ^ { 3 } + 1 }$ at x = -1. Enter just a reduced fraction.

If y = $\sqrt { u }$ and u = $x ^ { 3 }$ - 5 $x ^ { 2 }$ + 1, find $\frac { d y } { d x }$ .

Compute $\frac { \mathrm { dy } } { \mathrm { dx } }$ y = 4 $u ^ { 2 }$ + 8u + 4 and u = 3x + 1. Enter your answer as just a polynomial in x in standard form (no label).

The Derivative

Applications of the Derivative

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 3: Techniques of Differentiation

If f(x) = x (x−1)5( x - 1 ) ^ { 5 }(x−1)5 and g(x) = x2x ^ { 2 }x2 , find ddx\frac { \mathrm { d } } { \mathrm { dx } }dxd​ f(g(x)).

Differentiate -F(r) = r2\mathrm { r } ^ { 2 }r2 (r - 1)(r + 1 )−1) ^ { - 1 })−1

Assume 4 x3x ^ { 3 }x3 + 2xy - y3y ^ { 3 }y3 = 12\frac { 1 } { 2 }21​ . What is the slope of the graph at the point (12,−1)\left( \frac { 1 } { 2 } , - 1 \right)(21​,−1) ?

Let g(x) = x\sqrt { x }x​ . Using the chain rule, find an expression for the derivative of [g(f(x))].

Differentiate -f(x) = ( x2x ^ { 2 }x2 + 1 )3) ^ { 3 })3 ( x3x ^ { 3 }x3 - 1 )2)^2)2 at x = -1 Enter just an integer.

Use the chain rule to compute the derivative of (3x+1)3(3x+1)3+1\frac { ( 3 x + 1 ) ^ { 3 } } { ( 3 x + 1 ) ^ { 3 } + 1 }(3x+1)3+1(3x+1)3​ at x = -1. Enter just a reduced fraction.

If y = u\sqrt { u }u​ and u = x3x ^ { 3 }x3 - 5 x2x ^ { 2 }x2 + 1, find dydx\frac { d y } { d x }dxdy​ .

Compute dydx\frac { \mathrm { dy } } { \mathrm { dx } }dxdy​ y = 4 u2u ^ { 2 }u2 + 8u + 4 and u = 3x + 1. Enter your answer as just a polynomial in x in standard form (no label).

The Derivative

Applications of the Derivative

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

If f(x) = x $( x - 1 ) ^ { 5 }$ and g(x) = $x ^ { 2 }$ , find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)).

Differentiate -F(r) = $\mathrm { r } ^ { 2 }$ (r - 1)(r + 1 $) ^ { - 1 }$

Assume 4 $x ^ { 3 }$ + 2xy - $y ^ { 3 }$ = $\frac { 1 } { 2 }$ . What is the slope of the graph at the point $\left( \frac { 1 } { 2 } , - 1 \right)$ ?

Let g(x) = $\sqrt { x }$ . Using the chain rule, find an expression for the derivative of [g(f(x))].

Differentiate -f(x) = ( $x ^ { 2 }$ + 1 $) ^ { 3 }$ ( $x ^ { 3 }$ - 1 $)^2$ at x = -1 Enter just an integer.

Use the chain rule to compute the derivative of $\frac { ( 3 x + 1 ) ^ { 3 } } { ( 3 x + 1 ) ^ { 3 } + 1 }$ at x = -1. Enter just a reduced fraction.

If y = $\sqrt { u }$ and u = $x ^ { 3 }$ - 5 $x ^ { 2 }$ + 1, find $\frac { d y } { d x }$ .

Compute $\frac { \mathrm { dy } } { \mathrm { dx } }$ y = 4 $u ^ { 2 }$ + 8u + 4 and u = 3x + 1. Enter your answer as just a polynomial in x in standard form (no label).