What is the dervative of ((x+sin(2x))/(x-sin(2x)))^7?
use the power rule and the quotient rule, then the chain rule. what do you get?
13(x-sin(2x))(1-cos(2x))(x+sin(2x))^6-2(x+sin(2x))(1+cos(2x))/(x-sin(2x))^3
I mean 14 not 13
To find the derivative of the given function, we can use the quotient rule. The quotient rule states that if we have a function in the form of (f(x)/g(x)) raised to some power, we can find its derivative by following these steps:
1. Let f(x) = (x + sin(2x)), and g(x) = (x - sin(2x)).
2. Apply the quotient rule:
(f(x)/g(x))^n = ((g(x) * f'(x)) - (f(x) * g'(x)))/(g(x))^2
where f'(x) represents the derivative of f(x) and g'(x) represents the derivative of g(x).
3. Find the derivatives of f(x) and g(x).
- The derivative of f(x) = (x + sin(2x)) is f'(x) = 1 + 2cos(2x).
- The derivative of g(x) = (x - sin(2x)) is g'(x) = 1 - 2cos(2x).
4. Substitute the derivatives into the quotient rule formula:
((x + sin(2x))/(x - sin(2x)))^7 = ((x - sin(2x))(1 + 2cos(2x)) - (x + sin(2x))(1 - 2cos(2x)))/((x - sin(2x)))^2
5. Simplify the expression if necessary.
So, the derivative of ((x + sin(2x))/(x - sin(2x)))^7 is:
((x - sin(2x))(1 + 2cos(2x)) - (x + sin(2x))(1 - 2cos(2x)))/((x - sin(2x)))^2