hello, I'm trying to find the derivative of:
y= 1+cos (x) / sin (x)
thank you! :)
Hello! To find the derivative of the function y = (1 + cos(x))/sin(x), we can use the quotient rule. The quotient rule states that if we have a function in the form f(x)/g(x), the derivative can be calculated using the following formula:
(dy/dx) = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2
Let's break down the steps of finding the derivative of y = (1 + cos(x))/sin(x):
Step 1: Identify f(x) and g(x)
In this case, f(x) = (1 + cos(x)) and g(x) = sin(x).
Step 2: Find f'(x) and g'(x)
To find f'(x), we need to derive f(x) with respect to x. The derivative of (1 + cos(x)) is -sin(x). Similarly, the derivative of sin(x) is cos(x). Therefore, f'(x) = -sin(x) and g'(x) = cos(x).
Step 3: Apply the quotient rule formula
Now we can plug these values into the quotient rule formula:
(dy/dx) = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2
(dy/dx) = (sin(x) * (-sin(x)) - (1 + cos(x)) * cos(x)) / (sin(x))^2
Step 4: Simplify the expression
We can simplify the expression by expanding and simplifying:
(dy/dx) = (-sin^2(x) - cos(x) + cos^2(x)) / sin^2(x)
Step 5: Combine like terms
Combine like terms in the numerator:
(dy/dx) = (cos^2(x) - sin^2(x) - cos(x)) / sin^2(x)
This is the final simplified expression for the derivative of y = (1 + cos(x))/sin(x).
I hope this helps! Let me know if you have any further questions.