Question

Find f '(x) for f of x equals the square root of the quantity sine of 8 times x.
a. f prime of x equals the quotient of the cosine of 8 times x and 2 times the square root of the quantity sine of 8 times x
b. f prime of x equals the quotient of 4 times the cosine of 8 times x and the square root of the quantity sine of 8 times x
c. f prime of x equals the quotient of 4 and the square root of the quantity sine of 8 times x
d. f'(x) = sin(4x)

Answer 1

Can you please help? @DanJS @FibonacciChick666 @tkhunny

Answer 2

It is time for you to take the first shot at it.

\(f(x) = \sqrt{\sin(8x)}\) -- Go!

Answer 3

Aryana: You'll need to analyze this function to determine what types of mathematical operations are present and in which order.
The very first op is the Sqrt function.

Inside the Sqrt function is the expression sin 8x.

You must apply the Chain Rule here, and apply it twice.

Which function do you differentiate first?

Second?

Third?

Please show your stuff. Then either tkhunny or I would be happy to step in with any needed guidance.

Answer 4

Hint: The Sqrt function can also be written as a power function: y = x ^ (1/2), where the (1/2) is the exponent of x.

Answer 5

Please review "Power Rule with Chain Rule."

Answer 6

This is the dervative 4csc(8x)12cos(8x)

Answer 7

How so? Please show your work.

Find f '(x) for f of x equals the square root of the quantity sine of 8 times x.\[\frac{ d }{ dx }\sqrt{\sin 8x}=?\]

Answer 8

This is the same as \[\frac{ d }{ dx }{\(\sin 8x})^{1/2}\]

Answer 9

Use the power rule first, followed by the chain rule.

Answer 10

once you get the patterns down, the compound functions arent that bad

(sin(8x))^(1/2), start with the outside function first,

1/2 * sin(8x)^(-1/2) * (derivative of the next inside function sin(8x))

same thing continuing, cos(8x) * derivative of inside 8x
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\[\large \frac{ 1 }{ 2 }*[\sin(8x)]^{-1/2} * \cos(8x) * 8\]

Answer 11

the root is a function of variables X and Angle

the sin is a function of the Angle

the angle is a function of X
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to find the derivative of the root function w.r.t. X, you need to work thorugh the hierarchy

\[\frac{ df }{ dx } = \frac{ df }{ d \theta }*\frac{ d \theta }{ dx }\]

Answer 12

Aryana: Your response, please?