csc(x) derivative
The derivative of the cosecant function, csc(x), can be found using the chain rule in calculus
The derivative of the cosecant function, csc(x), can be found using the chain rule in calculus.
First, let’s define the cosecant function:
The cosecant function, csc(x), is defined as the reciprocal of the sine function, sin(x), where x is the input angle. Mathematically, we can express it as csc(x) = 1/sin(x). The cosecant function returns the ratio of the length of the hypotenuse to the length of the opposite side in a right triangle when the input angle is x.
Now, let’s find the derivative of csc(x):
To find the derivative, we can rewrite csc(x) as 1/sin(x). Let u(x) = sin(x), then we have csc(x) = 1/u(x). Now, applying the quotient rule, we get:
d/dx (csc(x)) = (u'(x) * v(x) – u(x) * v'(x))/[v(x)]^2
where u(x) = sin(x), v(x) = 1, u'(x) = d/dx(sin(x)), and v'(x) = d/dx(1).
The derivative of sin(x) is cos(x), so u'(x) = cos(x). The derivative of a constant is 0, so v'(x) = 0.
Substituting these values into the quotient rule, we have:
d/dx (csc(x)) = (cos(x) * 1 – sin(x) * 0)/[1]^2
Simplifying further, we get:
d/dx (csc(x)) = cos(x)/1 = cos(x)
Therefore, the derivative of the cosecant function, csc(x), is equal to the cosine function, cos(x).
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