d/dx sec x
sec x tan x
To find the derivative of sec x with respect to x, we can use the quotient rule of differentiation. Recall that:
(f/g)’ = (f’g – fg’)/g^2
For sec x, we have:
f(x) = 1
g(x) = cos x
Then, we can find the derivatives of f(x) and g(x):
f'(x) = 0
g'(x) = -sin x
Substituting into the quotient rule, we get:
(sec x)’ = {(0)(cos x) – (1)(-sin x)}/(cos x)^2
Simplifying the numerator and denominator, we have:
(sec x)’ = sin x/cos^2 x
Recall that tan x = sin x/cos x. Therefore, we can rewrite the derivative as:
(sec x)’ = tan x sec x
So, the derivative of sec x with respect to x is equal to tan x multiplied by sec x.
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