derivative of csc x
To find the derivative of csc(x), we will use the definition of the derivative and apply the chain rule
To find the derivative of csc(x), we will use the definition of the derivative and apply the chain rule.
The function csc(x) is defined as the reciprocal of the sine function, so we can rewrite it as:
csc(x) = 1 / sin(x)
To find the derivative, we will use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative of f(x) is given by:
f'(x) = (h(x) * g'(x) – g(x) * h'(x)) / (h(x))^2
In this case, g(x) = 1 and h(x) = sin(x).
Now, let’s find the derivative of g(x):
g'(x) = 0 (since it is a constant)
Next, let’s find the derivative of h(x):
h'(x) = cos(x)
Now we can use the quotient rule to find the derivative of csc(x):
csc'(x) = (sin(x) * 0 – 1 * cos(x)) / (sin(x))^2
= -cos(x) / (sin(x))^2
Therefore, the derivative of csc(x) is -cos(x) / (sin(x))^2.
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