d/dx sec(x)
To find the derivative of sec(x) with respect to x, we can use the quotient rule
To find the derivative of sec(x) with respect to x, we can use the quotient rule.
The quotient rule states that if we have a function f(x) = g(x) / h(x), then its derivative is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
For sec(x), we can express it as 1 / cos(x). Therefore, g(x) = 1 and h(x) = cos(x).
Now, let’s find the derivatives of g(x) and h(x):
g'(x) = 0 (since g(x) is a constant)
h'(x) = -sin(x) (using the derivative of cos(x) = -sin(x))
Now, let’s substitute these values into the quotient rule formula:
f'(x) = (0 * cos(x) – 1 * (-sin(x))) / (cos(x))^2
Simplifying further:
f'(x) = sin(x) / (cos(x))^2
Recall that the identity sec(x) = 1 / cos(x), so we can replace f'(x) in terms of sec(x):
d/dx sec(x) = sin(x) / (cos(x))^2
So, the derivative of sec(x) with respect to x is sin(x) / (cos(x))^2.
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