If y = (cos²x)/(1 + sin x), then dy/dx = ?
To find the derivative of y with respect to x, dy/dx, we will 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
Let's apply the quotient rule to find the derivative of y = (cos²x)/(1 + sin x):
First, we need to find g(x) and h(x):
g(x) = cos²x
h(x) = 1 + sin x
Next, we find the derivatives of g(x) and h(x):
g'(x) = d/dx (cos²x)
= 2cos x * (-sin x)
= -2cos x * sin x
h'(x) = d/dx (1 + sin x)
= 0 + cos x
= cos x
Now, apply the quotient rule:
dy/dx = (g'(x) * h(x) - g(x) * h'(x))/(h(x))^2
= (-2cos x * sin x * (1 + sin x) - cos²x * cos x)/(1 + sin x)^2
= (-2cos x * sin x - cos³x)/(1 + sin x)^2
So, dy/dx = (-2cos x * sin x - cos³x)/(1 + sin x)^2.
That is the derivative of y with respect to x.