d/dx[(cosx)^-1]
To find the derivative of the function f(x) = (cosx)^-1, we will use the chain rule
To find the derivative of the function f(x) = (cosx)^-1, we will use the chain rule.
The chain rule states that if we have a composite function g(f(x)), then the derivative of g(f(x)) is given by g'(f(x)) multiplied by f'(x). In this case, g(u) = u^-1 and f(x) = cosx.
Let’s differentiate the function step by step:
First, we need to find f'(x), which is the derivative of the inner function f(x) = cosx. This is given by f'(x) = -sinx.
Next, we differentiate g(u) = u^-1 with respect to u. Using the power rule of differentiation, we have:
g'(u) = -(1/u^2).
Now, by applying the chain rule and multiplying g'(f(x)) by f'(x), we get:
(d/dx)(cosx)^-1 = -(1/(cosx)^2) * (-sinx)
Simplifying this further, we have:
(d/dx)(cosx)^-1 = sinx / (cosx)^2
Therefore, the derivative of the function f(x) = (cosx)^-1 is sinx / (cosx)^2.
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