Derivative of csc(x): Step-by-Step Guide and Application of the Chain Rule

d/dx(cscx)

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

To find the derivative of csc(x) with respect to x, we need to 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, f(x) = csc(x) = 1/sin(x).

So, we can rewrite the function as f(g(x)) = (1/g(x)), where g(x) = sin(x).

Now, let’s find the derivatives of f(x) and g(x):

Derivative of f(x):
f'(x) = d/dx(1/sin(x))

Using the quotient rule, we have:
f'(x) = (sin(x)(0) – 1(cos(x)))/(sin(x))^2

Simplifying further:
f'(x) = -cos(x)/(sin(x))^2

Derivative of g(x):
g'(x) = d/dx(sin(x)) = cos(x)

Now, applying the chain rule, we can find the derivative of csc(x):

d/dx(csc(x)) = f'(g(x)) * g'(x)
= -cos(x)/(sin(x))^2 * cos(x)
= -cos^2(x)/(sin^2(x))
= -cot^2(x)

So, the derivative of csc(x) with respect to x is -cot^2(x).

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