How to Find the Derivative of csc(x) Using the Quotient Rule and Reciprocal Identity

d/dx (csc x)

To find the derivative of the function f(x) = csc(x), we can use the quotient rule or reciprocal identity

To find the derivative of the function f(x) = csc(x), we can use the quotient rule or reciprocal identity.

1. Using the quotient rule:
The quotient rule states that if f(x) = g(x)/h(x), then the derivative of f(x) can be found using the formula:
f'(x) = (g'(x)h(x) – g(x)h'(x))/[h(x)]^2

In this case, g(x) = 1 and h(x) = sin(x).
So, g'(x) = 0 (derivative of a constant is zero)
And h'(x) = cos(x) (derivative of sin(x) is cos(x))

Therefore, applying the quotient rule:
f'(x) = (0*cos(x) – 1*cos(x))/[sin(x)]^2
= -cos(x)/[sin(x)]^2

So, d/dx (csc x) = -cos(x)/[sin(x)]^2

2. Using the reciprocal identity:
Another way to find the derivative of csc(x) is by using the reciprocal identity:
csc(x) = 1/sin(x)

Now we can rewrite csc(x) as:
f(x) = (sin(x))^(-1)

To find the derivative of f(x):

f'(x) = d/dx [(sin(x))^(-1)]

We can use the chain rule to differentiate (sin(x))^(-1):

f'(x) = -[(sin(x))^(-2)] * cos(x)

Now, substituting back csc(x) for f(x):

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

Therefore, both the quotient rule and the reciprocal identity give us the same derivative:

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

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