Derivative of csc(x) with Respect to x | Step-by-Step Guide and Solution

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.

The chain rule states that if we have a function f(g(x)), where f(u) and g(x) are both differentiable functions, then the derivative of f(g(x)) with respect to x is given by:

df/dx = df/du * du/dx

In this case, f(u) = csc(u) and g(x) = x. So, let’s break down the problem step by step:

Step 1: Find the derivative of f(u) = csc(u)

To find the derivative of csc(u), we can rewrite it in terms of sine:

csc(u) = 1 / sin(u)

Now, using the quotient rule, we can differentiate the right-hand side:

d/dx (1 / sin(u)) = (0 * sin(u) – 1 * cos(u)) / (sin^2(u))

Simplifying further:

– cos(u) / sin^2(u) = -cot(u) / sin(u)

So, the derivative of f(u) = csc(u) is f'(u) = -cot(u) / sin(u).

Step 2: Find the derivative of g(x) = x

The derivative of x with respect to x is simply 1.

Step 3: Apply the chain rule

Using the chain rule, we have:

d/dx (csc(x)) = f'(g(x)) * g'(x)

Substituting the derivatives we found in Step 1 and Step 2:

d/dx (csc(x)) = (-cot(g(x)) / sin(g(x))) * 1

Since g(x) = x, we have:

d/dx (csc(x)) = -cot(x) / sin(x)

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

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