d/dx(cscx)
To find the derivative of cscx, we can use the quotient rule
To find the derivative of cscx, we can 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) with respect to x is given by:
f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
Applying this rule to the function f(x) = cscx, we have g(x) = 1 and h(x) = sinx. Therefore, g'(x) = 0 (since the derivative of a constant is always 0) and h'(x) = cosx (the derivative of sinx is cosx).
Now let’s substitute these values into the quotient rule formula:
f'(x) = [0 * sinx – 1 * cosx] / [sinx]^2
Simplifying, we get:
f'(x) = -cosx / sin^2x
Since cscx is equal to 1/sinx, we can write the derivative as:
d/dx(cscx) = -cotx / sinx
Therefore, the derivative of cscx with respect to x is -cotx / sinx.
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