find the differentiate y=cscx/�ãx
To find the derivative of y = csc(x) / √x, we need to follow these steps:
Step 1: Rewrite the equation using trigonometric identities if necessary.
In this case, csc(x) is the reciprocal of sin(x), so we can rewrite the equation as y = 1/sin(x) / √x.
Step 2: Simplify the equation.
To simplify, rewrite the equation as y = √x / sin(x). We want to find the derivative of this expression.
Step 3: Use the Quotient Rule.
The Quotient Rule states that if we have a function f(x) divided by g(x), the derivative of the function can be found by:
dy/dx = (g(x) * f'(x) - f(x) * g'(x)) / [g(x)]^2
In this case, f(x) = √x and g(x) = sin(x). To find f'(x) and g'(x), we need to differentiate both functions.
Step 4: Differentiate f(x) and g(x).
To differentiate f(x) = √x, we can use the Power Rule, which states that if we have a function of the form f(x) = x^n, the derivative is given by:
f'(x) = n * x^(n-1)
By applying the Power Rule to √x, we get f'(x) = (1/2) * x^(-1/2).
Differentiating g(x) = sin(x) can be done using the derivative of a trigonometric function, which states that the derivative of sin(x) is cos(x).
So, f'(x) = (1/2) * x^(-1/2) and g'(x) = cos(x).
Step 5: Apply the Quotient Rule to find the derivative of y.
Using the Quotient Rule, we have:
dy/dx = (g(x) * f'(x) - f(x) * g'(x)) / [g(x)]^2
Plugging in the values, we get:
dy/dx = (sin(x) * (1/2) * x^(-1/2) - √x * cos(x)) / [sin(x)]^2
Simplifying the expression, we have:
dy/dx = (sin(x) - 2√x * cos(x)) / (2x * sin^2(x))
Therefore, the derivative of y = csc(x) / √x is given by:
dy/dx = (sin(x) - 2√x * cos(x)) / (2x * sin^2(x))