d/dx sec x
sec x tan x
We can use the formula for the derivative of the secant function, which is:
d/dx sec x = sec x tan x
Therefore, the derivative of sec x is equal to sec x multiplied by the tangent of x.
To understand how this formula is derived, we can start with the definition of the secant function:
sec x = 1/cos x
Then, we can use the quotient rule to find its derivative:
d/dx sec x = [cos x (d/dx 1) – 1 (d/dx cos x)] / cos²x
Since d/dx 1 = 0 and d/dx cos x = -sin x, we get:
d/dx sec x = -sin x / cos²x
Next, we can use the trigonometric identity:
tan x = sin x / cos x
to simplify the expression:
d/dx sec x = -tan x / cos x
Finally, we can multiply both the numerator and denominator by sec x to obtain the formula we started with:
d/dx sec x = sec x tan x
Therefore, the derivative of sec x is equal to sec x multiplied by the tangent of x.
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