To find the derivative (differential) of the function y = -2cos(3x), we can follow these steps:
Step 1: Understand the problem.
We're given the function y = -2cos(3x) and asked to find the derivative.
Step 2: Recognize that the function is a compound function.
The function y = -2cos(3x) is a compound function because it's a combination of two simpler functions: -2cos(x) and 3x.
Step 3: Use the chain rule.
The chain rule states that the derivative of a composition of functions is the product of the derivative of the inside function and the outer function. Precisely if y = f(g(x)) then dy/dx = f'(g(x))g'(x).
The outer function here is -2cos(x) and the inner function is 3x.
Step 4: Differentiate the outer function.
The derivative of -2cos(x) with respect to x is 2sin(x).
Step 5: Differentiate the inner function.
The derivative of 3x with respect to x is 3.
Step 6: Apply the chain rule.
According to the chain rule, we need to multiply the derivative of the outer function by the derivative of the inner function. That gives us 2sin(x)*3 which simplifies to 6sin(x).
Step 7: But recall that in step 4, we have -2sin(x) not 2sin(x), also from the chain rule, we remember that in f'(g(x))g'(x) we are replacing every x in f'(x) by g(x). Hence, the correct derivative of -2cos(3x) is 6sin(3x) rather than 6sin(x).
Thus, the derivative dy/dx of the function y = -2cos(3x) is 6sin(3x).