Simplified: Finding the Derivative of csc(x) Using the Quotient Rule in Calculus

𝑑/𝑑𝑥[csc 𝑥]

To find the derivative of csc(x), we can use the quotient rule in calculus

To find the derivative of csc(x), we can use the quotient rule in calculus.

The quotient rule states that if we have a function f(x) = g(x) / h(x), where g(x) and h(x) are both functions of 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

In our case, g(x) = 1 and h(x) = sin(x) because csc(x) = 1/sin(x).

Using the quotient rule, we can find the derivative of csc(x) with respect to x.

First, let’s find g'(x) and h'(x):

g'(x) = 0 (since g(x) = 1 is a constant)

h'(x) = cos(x) (by differentiating sin(x) using the chain rule)

Now, applying the quotient rule, we have:

csc'(x) = [g'(x) * h(x) – g(x) * h'(x)] / (h(x))^2
= [0 * sin(x) – 1 * cos(x)] / (sin(x))^2
= -cos(x) / sin^2(x)

Simplifying further, we can rewrite the expression as:

csc'(x) = -cos(x) / sin^2(x)
= -1 / (sin(x) * sin(x))
= -1 / sin^2(x)

Therefore, the derivative of csc(x) with respect to x is -1 / sin^2(x).

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