Dx {csc (x)}=?
To find the derivative of csc(x), we can start by expressing it as a reciprocal of sine function:
csc(x) = 1/sin(x)
Now, let’s find the derivative using the quotient rule:
f(x) = 1 and g(x) = sin(x)
f'(x) = 0 and g'(x) = cos(x)
Using the quotient rule, the derivative of csc(x) is given by:
(csc(x))’ = (f'(x)g(x) – f(x)g'(x))/(g(x))^2
= (0 * sin(x) – 1 * cos(x))/(sin(x))^2
= -cos(x)/(sin(x))^2
Alternatively, we can rewrite csc(x) as:
csc(x) = (1/sin(x)) * (sin(x)/sin(x))
= (sin(x))^(-1) * (sin(x)/sin(x))
= (sin(x))^(-1)
Taking the derivative using the chain rule, we have:
(csc(x))’ = (-1)(sin(x))^(-2) * cos(x)
= -cos(x)/(sin(x))^2
Therefore, the derivative of csc(x) is -cos(x)/(sin(x))^2
To find the derivative of csc(x), we can start by expressing it as a reciprocal of sine function:
csc(x) = 1/sin(x)
Now, let’s find the derivative using the quotient rule:
f(x) = 1 and g(x) = sin(x)
f'(x) = 0 and g'(x) = cos(x)
Using the quotient rule, the derivative of csc(x) is given by:
(csc(x))’ = (f'(x)g(x) – f(x)g'(x))/(g(x))^2
= (0 * sin(x) – 1 * cos(x))/(sin(x))^2
= -cos(x)/(sin(x))^2
Alternatively, we can rewrite csc(x) as:
csc(x) = (1/sin(x)) * (sin(x)/sin(x))
= (sin(x))^(-1) * (sin(x)/sin(x))
= (sin(x))^(-1)
Taking the derivative using the chain rule, we have:
(csc(x))’ = (-1)(sin(x))^(-2) * cos(x)
= -cos(x)/(sin(x))^2
Therefore, the derivative of csc(x) is -cos(x)/(sin(x))^2.
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