To find the derivative of a sin(2x) function, you must be familiar with derivatives of trigonometric functions and the chain rule for finding derivatives. You need scratch paper and can use a graphing calculator to check coordinates and slopes at specific values.
- Use the chain rule
The chain rule provides a method for taking the derivative of a function in which one operation happens within another. In function f(x) = sin(2x), the operation 2x happens within the sine function. If g(x) = sin(x) and h(x) = 2x, then g(h(x)) = sin(2x) = f(x). The chain rule lets you take the derivative of the outside and multiply it by the derivative of the inside. The chain rule states that the derivative of g(h(x)) = g'(h(x))*h'(x). Therefore, f'(x) =(d/dx)*sin(2x) = (d*sin(2x)/dx)*(d*2x/dx).
- Derive the sine function
The derivative of a sine function is a cosine. This is true because both functions are periodical functions with the same period length, but the cosine function is at value 0 when the slope of the sine function is equal to 0. f'(x) = (d*sin(2x)/dx)*(d*2x/dx) = cos(2x)*(d*2x/dx)
- Derive the function in the parentheses
The derivative of any constant multiplied by x to the first power is that coefficient. d*2x/dx = 2 Therefore, f'(x) = cos(2x)*(d*2x/dx) = cos(2x)*2 f'(x) = 2cos(2x)