d/dx(csc(x))
To find the derivative of csc(x) with respect to x, we can use the chain rule and the fact that csc(x) is equal to 1/sin(x)
To find the derivative of csc(x) with respect to x, we can use the chain rule and the fact that csc(x) is equal to 1/sin(x).
Let’s start by writing csc(x) as 1/sin(x):
csc(x) = 1/sin(x)
Now, let’s use the quotient rule to find the derivative:
d/dx(csc(x)) = d/dx(1/sin(x))
Using the quotient rule, the derivative of 1/sin(x) with respect to x is given by:
= (sin(x)(d/dx(1)) – 1(d/dx(sin(x)))) / (sin^2(x))
Now, let’s calculate the derivatives:
The derivative of 1 with respect to x is 0, because 1 is a constant.
The derivative of sin(x) with respect to x is cos(x), according to the chain rule.
Substituting these derivative values into the quotient rule formula, we get:
= (sin(x)(0) – 1(cos(x))) / (sin^2(x))
Simplifying further, we have:
= -cos(x) / (sin^2(x))
So, the derivative of csc(x) with respect to x is equal to -cos(x) divided by the square of sin(x).
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