Calculating the Derivative of csc(x) using the Chain Rule | Simplified Proof and Solution

d/dx cscx

To find the derivative of cscx with respect to x (denoted as d/dx cscx), we can use the chain rule of differentiation

To find the derivative of cscx with respect to x (denoted as d/dx cscx), we can use the chain rule of differentiation. The derivative of any function f(g(x)) is given by f'(g(x)) * g'(x).

First, let’s express cscx in terms of sine: cscx = 1/sinx.

Now, applying the chain rule, we have:

d/dx cscx = d/dx (1/sinx)

Let’s find the derivative of sinx first. The derivative of sinx with respect to x is cosx, so sinx has derivative of cosx.

Now, let’s find the derivative of 1/sinx. Using the quotient rule, we have:

d/dx (1/sinx) = (sinx * d/dx(1) – 1 * d/dx(sinx))/(sinx)^2

Since d/dx(1) = 0, and d/dx(sinx) = cosx, we can simplify further:

d/dx (1/sinx) = -cosx/(sinx)^2

Therefore, the derivative of cscx with respect to x (d/dx cscx) is equal to -cosx/(sinx)^2.

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