Which of the following functions does not have a derivative of zero?

A. y=-1/100
B. y=4(pi)^2-9
C. y=sin^2x+cos^2x
D. y=cscxtanxcosx
E. All do

well, which function is not constant?

- 1 / 100 = const.

4 ( pi ) ^ 2 - 9 = const.

sin ^ 2 x + cos ^ 2 x = 1 = const.

csc x tan x cosx = ( 1 / sin x ) * ( sin x / cos x ) * cos x = 1 = const.

The derivative of a constant is zero.

Answer :

E. All do

To determine which function does not have a derivative of zero, we need to find the derivative of each function and check if it is equal to zero.

Let's find the derivatives of each function:

A. y = -1/100
The derivative of a constant is zero. So, the derivative of y = -1/100 is zero.

B. y = 4(pi)^2 - 9
The derivative of a constant is zero. So, the derivative of y = 4(pi)^2 -9 is zero.

C. y = sin^2(x) + cos^2(x)
The derivative of sin^2(x) is 2sin(x)cos(x), and the derivative of cos^2(x) is -2sin(x)cos(x). Adding these derivatives together, we get 2sin(x)cos(x) - 2sin(x)cos(x) = 0.

D. y = csc(x)tan(x)cos(x)
To find the derivative of this function, we can use the product rule.
The derivative of csc(x) is -csc(x)cot(x) and the derivative of tan(x) is sec^2(x). The derivative of cos(x) is -sin(x).
Using the product rule, we have:
dy/dx = -csc(x)cot(x)cos(x) + csc(x)sec^2(x)cos(x) - csc(x)tan(x)sin(x)
Simplifying, dy/dx = -csc(x)[cot(x)cos(x) - sec^2(x)cos(x) + tan(x)sin(x)]
Since the expression inside the parentheses can be simplified to zero by using trigonometric identities, we can conclude that the derivative dy/dx is zero.

E. All the functions have a derivative of zero because the derivative of a constant, a trigonometric identity, or the product of trigonometric functions that can be simplified to zero, is always zero.

Therefore, the answer is E. All the given functions have a derivative of zero.

To determine which of the functions does not have a derivative of zero, we need to find the derivative of each function and check if it is equal to zero.

A. y = -1/100
To find the derivative, we differentiate the function with respect to x:
dy/dx = 0
Since the derivative is zero, option A has a derivative of zero.

B. y = 4(pi)^2 - 9
Again, let's find the derivative:
dy/dx = 0
The derivative is also zero for option B.

C. y = sin^2x + cos^2x
Differentiating with respect to x:
dy/dx = 2sin(x)cos(x) - 2sin(x)cos(x)
Simplifying:
dy/dx = 0
The derivative of option C is zero as well.

D. y = csc(x)tan(x)cos(x)
Differentiating:
dy/dx = -cosec(x)cot(x)sin(x) + sec(x)sec^2(x)cos(x) - sec(x)tan(x)sin(x)
Simplifying:
dy/dx = -cosec(x)cot(x)sin(x) + sec^3(x)cos(x) - sec(x)tan(x)sin(x)
The derivative is not equal to zero, so option D does not have a derivative of zero.

E. All do
Based on the above analysis, we can see that all the options have a derivative of zero except for option D. Therefore, the answer is option D, y = csc(x)tan(x)cos(x), which does not have a derivative of zero.