Question 1

g(x) = x⁴f(x). Find g'(4), given that f(4) = 3 and f '(4) = 6.

Accepted Answer

Using the product rule for differentiation, g'(x) = 4x^3f(x) + x^4f'(x). Substituting x = 4, f(4) = 3, and f'(4) = 6, we get g'(4) = 4(4^3)(3) + (4^4)(6) = 2,304.

Question 2

Find f '(u) if  .

Accepted Answer

The answer of Find f '(u) if  ....

Question 3

Find f '(x) if  .

Accepted Answer

The answer of Find f '(x) if  ....

Question 4

If f(3) = 4, f '(3) = -5, g(3) = 2, g '(3) = -4, find F'(3) if F(x) = 2f(x) - g(x) .

Accepted Answer

The answer of If f(3) = 4, f '(3) =...

Question 5

If  , find  .

Accepted Answer

When we look at the two given values, the only difference is in the fourth block. The first value has 99746d078d5f and the second value has 27a5ee5af53a. If we compare the two values in binary, we can see that the only difference is in the 13th bit from the right (marked as bold below):

99746d078d5f: 1001 1001 0111 0100 0110 1101 0000 0111 1000 1101 0101 1111
27a5ee5af53a: 0010 0111 1010 0101 1110 1110 0101 1010 1111 0101 0011 1010

In the second value, the 13th bit from the right is 1 while in the first value it is 0. Thus, to get the second value, we only need to change this bit from 0 to 1. Looking at the answer choices, we can see that the only one with this change is B) 11ead0bc_8d6d_e513_99a0_1793f2fe7908_TB6988_11.

Question 6

Find if y = (x² - 1)(x³ + 3x).

Accepted Answer

The answer of Find if y = (x² -...

Question 7

Answer true or false. If  ,  .

Accepted Answer

The two values are different in the third segment (8d70_f27e vs 8d70_f27f), so they are not equal.

Question 8

Find f '(x) if f(x) = (x² + 5)(x³ - 4x² + x).

Accepted Answer

The answer of Find f '(x) if f(x) = (x²...

Question 9

If f(4) = 7, f '(4) = 2, g(4) = 2, g '(4) = -2, find F'(4) if F(x) =  .

Accepted Answer

The answer of If f(4) = 7, f '(4) =...

Question 10

Find f '(t) if  .

Accepted Answer

The answer of Find f '(t) if  ....

Question 11

y = 4x² + 9x + 2. Find d²y/dx².

Accepted Answer

The second derivative of y = 4x^2 + 9x + 2 is d^2y/dx^2 = 8.

Question 12

Find the equations of the tangent and normal to the graph of  at x = 4.

Accepted Answer

The answer of Find the equations of the tangent and...

Question 13

Find dy/dx if y = 5x⁸.

Accepted Answer

Taking the derivative of the given expression, we use the power rule and the chain rule: dy/dx = 5(⁸)(d/dx)(x) = 5(⁸) Since the derivative of x is 1, we can simplify this to: dy/dx = 5x(⁸) Using the product rule and the power rule for differentiation, we have: dy/dx = 5x(1)(U1)(P17)(1)(S1)(S1)(P0) + 5(1)(⁸)(1) Simplifying this gives: dy/dx = 5x⁷ + 5⁸ Which is equivalent to answer choice C.

Question 14

Find dy/dx if y = 3(x⁶ - 6x + 1).

Accepted Answer

To find dy/dx, first expand y = 3(x^6 - 6x + 1) to y = 3x^6 - 18x + 3. Then, differentiate term by term: dy/dx = 18x^5 - 18.

Question 15

If  ,

Accepted Answer

The only difference between the two given values is the third group of characters. In the first value, it is "3337" and in the second value, it is "5a48". By comparing the two, we can see that every digit in the second group is one more than the corresponding digit in the first group (3+1=4, 3+1=4, 3+1=4, 7+1=8). Therefore, in the third group, we should have "3337" + 1 = "3338", which is "815b" in hexadecimal. Thus, the correct answer is D: 11ead0bc_8d6e_815b_99a0_1f0577641f55_TB6988_11.

Question 16

Find  if y =  .

Accepted Answer

The answer of Find  if y =  ....

Question 17

If f(6) = 48, f '(6) = 5, g(6) = 4, g '(6) = -3, find F'(6) if F(x) =  .

Accepted Answer

The answer of If f(6) = 48, f '(6) =...

Question 18

Find dy/dx if  .

Accepted Answer

The given expression is a hexadecimal constant and does not contain any variables or functions with regards to which differentiation can be done. Hence, dy/dx is always equal to zero.

Question 19

If  , find  .

Accepted Answer

The answer of If  , find  .
...

Question 20

Answer true or false. If f, g, and h are differentiable functions, and h $\neq$ 0 anywhere on its domain, then  .

Accepted Answer

This statement is a direct application of the quotient rule of differentiation, which states that if f and g are differentiable functions and g is nonzero anywhere on its domain, then (f/g)' = (f'g - fg')/g^2. In this case, letting F = f/g, we have F' = (f'g - fg')/g^2, and since h is also differentiable and nonzero, we can differentiate again to obtain F'' = [f''g + 2f'g' - g''f]/g^3, which is also well-defined and nonzero. Thus, F has a nonzero second derivative, which implies that it is not constant.

Quiz 2: The Derivative

g(x) = x⁴f(x) . Find g'(4) , given that f(4) = 3 and f '(4) = 6.

Find f '(u) if .

Find f '(x) if .

If f(3) = 4, f '(3) = -5, g(3) = 2, g '(3) = -4, find F'(3) if F(x) = 2f(x) - g(x) .

If , find .

Find if y = (x² - 1)(x³ + 3x).

Answer true or false. If , .

Find f '(x) if f(x) = (x² + 5)(x³ - 4x² + x).

If f(4) = 7, f '(4) = 2, g(4) = 2, g '(4) = -2, find F'(4) if F(x) = .

Find f '(t) if .

y = 4x² + 9x + 2. Find d²y/dx².

Find the equations of the tangent and normal to the graph of at x = 4.

Find dy/dx if y = 5x⁸.

Find dy/dx if y = 3(x⁶ - 6x + 1) .

If ,

Find if y = .

If f(6) = 48, f '(6) = 5, g(6) = 4, g '(6) = -3, find F'(6) if F(x) = .

Find dy/dx if .

If , find .

Answer true or false. If f, g, and h are differentiable functions, and h $\neq$ 0 anywhere on its domain, then $Answer true or false. If f, g, and h are differentiable functions, and h \neq 0 anywhere on its domain, then .$ .

Quiz 2: The Derivative

g(x) = x4f(x) . Find g'(4) , given that f(4) = 3 and f '(4) = 6.

Find f '(u) if .

Find f '(x) if .

If f(3) = 4, f '(3) = -5, g(3) = 2, g '(3) = -4, find F'(3) if F(x) = 2f(x) - g(x) .

If , find .

Find if y = (x2 - 1)(x3 + 3x).

Answer true or false. If , .

Find f '(x) if f(x) = (x2 + 5)(x3 - 4x2 + x).

If f(4) = 7, f '(4) = 2, g(4) = 2, g '(4) = -2, find F'(4) if F(x) = .

Find f '(t) if .

y = 4x2 + 9x + 2. Find d2y/dx2.

Find the equations of the tangent and normal to the graph of at x = 4.

Find dy/dx if y = 5x8.

Find dy/dx if y = 3(x6 - 6x + 1) .

If ,

Find if y = .

If f(6) = 48, f '(6) = 5, g(6) = 4, g '(6) = -3, find F'(6) if F(x) = .

Find dy/dx if .

If , find .

Answer true or false. If f, g, and h are differentiable functions, and h ≠\neq= 0 anywhere on its domain, then .

g(x) = x⁴f(x) . Find g'(4) , given that f(4) = 3 and f '(4) = 6.

Find if y = (x² - 1)(x³ + 3x).

Find f '(x) if f(x) = (x² + 5)(x³ - 4x² + x).

y = 4x² + 9x + 2. Find d²y/dx².

Find dy/dx if y = 5x⁸.

Find dy/dx if y = 3(x⁶ - 6x + 1) .

Answer true or false. If f, g, and h are differentiable functions, and h $\neq$ 0 anywhere on its domain, then $Answer true or false. If f, g, and h are differentiable functions, and h \neq 0 anywhere on its domain, then .$ .