Let y=cos x+sin x/cos x -sin x. Find dy/dx
To find the derivative of y, which is dy/dx, we can use the quotient rule.
The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
So, to find dy/dx of y = (cos(x) + sin(x))/(cos(x) - sin(x)), let's differentiate the numerator and denominator separately:
1. Differentiating the numerator:
The derivative of cos(x) is -sin(x), and the derivative of sin(x) is cos(x). Hence, the derivative of cos(x) + sin(x) with respect to x is -sin(x) + cos(x).
2. Differentiating the denominator:
The derivative of cos(x) is -sin(x), and the derivative of -sin(x) is -cos(x). Hence, the derivative of cos(x) - sin(x) with respect to x is -sin(x) - cos(x).
Now, we can substitute these derivatives into the quotient rule:
dy/dx = ((-sin(x) + cos(x)) * (cos(x) - sin(x)) - (cos(x) + sin(x)) * (-sin(x) - cos(x))) / (cos(x) - sin(x))^2
Simplifying this expression will give you the final answer for dy/dx in terms of x.