How to find the derivative of the cosecant function (cscx) using the quotient rule

ddx cscx

To find the derivative of the cosecant function (cscx), let’s start by recalling the definition of the cosecant function:

csc(x) = 1 / sin(x)

Now, we will use the quotient rule to differentiate cscx

To find the derivative of the cosecant function (cscx), let’s start by recalling the definition of the cosecant function:

csc(x) = 1 / sin(x)

Now, we will use the quotient rule to differentiate cscx.

The quotient rule states that if we have a function u(x) = f(x) / g(x), then its derivative is given by:

du/dx = (g(x)f'(x) – f(x)g'(x)) / (g(x))^2

In the case of cscx, f(x) = 1 and g(x) = sin(x). Therefore, we need to find the derivatives of f(x) and g(x) to apply the quotient rule.

f'(x) = d/dx (1) = 0 (since the derivative of a constant is zero)

g'(x) = d/dx (sin(x)) = cos(x) (using the derivative of sin x, which is cos(x))

Now, we can substitute these values into the quotient rule to find the derivative of cscx:

d/dx (cscx) = (sin(x)(0) – 1(cos(x))) / (sin(x))^2

Simplifying further:

d/dx (cscx) = (- cos(x)) / (sin^2(x))

So, the derivative of cscx is equal to -cos(x) divided by the square of sin(x), or simply written as:

d/dx (cscx) = -cos(x) / sin^2(x)

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