Question

What is the derivative of the expression where 'e' is raised to the power of '6x' and it equals the 'sine' of the product of 'x' and '4y', concerning the variable 'x'?

Answer 1

The derivative of e raised to the power of 6x equal to the sine of the product of x and 4y, concerning x, is calculated using the chain rule and product rule. The final answer is 6e^{6x} = 4y.cos(4xy).

Step-by-step explanation:

The question is asking for the derivative of the function e6x = sin(4xy) with respect to x. To find this derivative, we will use the chain rule and the product rule of differentiation.

Steps for Differentiation

1. First, differentiate both sides of the equation with respect to x considering y as a constant. The derivative of e6x with respect to x is 6e6x.
2. On the right side, the derivative of sin(4xy) with respect to x is cos(4xy) multiplied by the derivative of (4xy) with respect to x. Since y is considered constant, the derivative of 4xy with respect to x is 4y.
3. So, the derivative of sin(4xy) concerning x is 4y.cos(4xy).
4. Setting both derivatives equal gives us 6e6x = 4y.cos(4xy), which is the derivative we were looking to find.