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) can be found using the formula:
f'(x) = (g'(x)*h(x) – g(x)*h'(x)) / (h(x))^2
In this case, our function is f(x) = cscx. We can rewrite cscx as 1/sinx, since cscx is equal to the reciprocal of the sine function.
f(x) = 1/sinx
Now, we can see that g(x) = 1 and h(x) = sinx. To find the derivative, we need to find g'(x) (the derivative of g(x)) and h'(x) (the derivative of h(x)).
g'(x) = 0, since the derivative of a constant (in this case, 1) is always zero.
h'(x) = cosx, since the derivative of sinx is cosx.
Now, we can substitute these values into the quotient rule formula:
f'(x) = (0*sinx – 1*cosx) / (sinx)^2
= -cosx / sin^2x
= -cotx*cscx
Therefore, the derivative of cscx is -cotx*cscx.
Remember that the cotx function represents the reciprocal of the tanx function, and the cscx function represents the reciprocal of the sinx function.
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