Finding the Derivative of csc(x) using the Chain Rule and Quotient Rule

d/dx[cscx]

To find the derivative of csc(x) with respect to x, we can use the chain rule

To find the derivative of csc(x) with respect to x, we can use the chain rule.

Recall that the derivative of a function f(g(x)) with respect to x is given by (f'(g(x)) * g'(x)).

In this case, we have csc(x) = 1/sin(x), so we can rewrite the derivative as:

d/dx[cscx] = d/dx[1/sin(x)]

Using the quotient rule, which states that the derivative of f(x)/g(x) is (f'(x) * g(x) – f(x) * g'(x)) / (g(x))^2, we can differentiate the numerator and denominator separately.

Considering the numerator, we have f(x) = 1 and g(x) = sin(x). Differentiating them, we get f'(x) = 0 and g'(x) = cos(x).

Now, applying the quotient rule, we have:

d/dx[1/sin(x)] = (0 * sin(x) – 1 * cos(x)) / (sin(x))^2

Simplifying this expression, we get:

d/dx[1/sin(x)] = -cos(x) / (sin(x))^2

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

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