The derivative of the function f(x) = sin(cos(3x)) with respect to x is found using the chain rule, which allows us to differentiate composite functions. Here’s how it’s done step by step:
The derivative of sin of any argument is the cosine of that argument. So, the outer derivative is the cosine of the inner function:
d/dx sin(cos(3x)) = cos(cos(3x)) * d/dx (cos(3x))
The derivative of cos(3x) with respect to x is -3sin(3x), because the derivative of cos is -sin and we must also apply the chain rule to the innermost function, 3x, multiplying by its derivative, 3:
d/dx (cos(3x)) = -sin(3x) * 3
Combining these results, we get:
d/dx sin(cos(3x)) = cos(cos(3x)) * (-sin(3x) * 3)
Which simplifies to:
d/dx sin(cos(3x)) = -3 cos(cos(3x)) sin(3x)