W=6x^4 SQRT4y ln (3z)

find F with regards to x
Is 24x^3 SQRT4y ln (3z) correct?

I don't understand the question.

What and where is F ?
Where does the square root end
is it √(4y) * ln(3z) , or
is it √(4yln(3z)) or
....
Are you finding the derivative of W with respect to x ?

(no regards in Calculus, usually we give our regards to Broadway)

W=6x^4 * SQRT4y * ln * (3z)

Yes, the derivative of W with respect to x

so you have 3 factors , assuming x, y and z are variables.

use the product rule

W = (6x^4)(4y)^.5 (ln (3z))
dW/dx = 24x^3((4y)^.5 (ln (3z)) + (6x^4)(ln(3z)) (2y dy/dx) + (6x^4)(4y)^.5 (1/z) dz/dx

the middle term should be

dW/dx = 24x^3((4y)^.5 (ln (3z)) + (6x^4)(ln(3z)) ((1/4)(4y)^-.5 dy/dx) + (6x^4)(4y)^.5 (1/z) dz/dx

thanks, that's pretty close to what I got

To find the derivative of F with respect to x, we need to differentiate each term of the expression separately.

Let's break down the given expression: W = 6x^4 SQRT(4y) ln (3z)

Step 1: Differentiate the term 6x^4.
The derivative of 6x^4 with respect to x is obtained by multiplying the current exponent (4) by the coefficient (6). So the derivative of 6x^4 with respect to x is 24x^3.

Step 2: Differentiate the term SQRT(4y).
To differentiate SQRT(4y) with respect to x, we treat 4y as a constant since it does not contain any x terms. So the derivative of SQRT(4y) with respect to x is 0.

Step 3: Differentiate the term ln(3z).
To differentiate ln(3z) with respect to x, we treat 3z as a constant since it does not contain any x terms. So the derivative of ln(3z) with respect to x is also 0.

Now, let's combine these results to find the derivative of W with respect to x:

F = 24x^3 * SQRT(4y) * ln(3z)

Therefore, the final derivative of W (F) with respect to x is indeed 24x^3 SQRT(4y) ln(3z).

So, 24x^3 SQRT(4y) ln(3z) is the correct answer.