d/dx(secx)
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 the derivative of f(x) can be found using the formula:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
Now, let’s apply this to our function f(x) = sec(x), where g(x) = 1 and h(x) = cos(x):
f'(x) = (1 * cos(x) – sec(x) * (-sin(x))) / (cos(x))^2
Since sec(x) is equal to 1/cos(x), we can simplify this to:
f'(x) = (cos(x) + sec(x) * sin(x)) / (cos(x))^2
To simplify this further, we can multiply the numerator and denominator by cos(x):
f'(x) = (cos(x)^2 + sec(x) * sin(x) * cos(x)) / (cos(x))^3
Applying the trigonometric identity cos(x)^2 = 1 – sin(x)^2, we get:
f'(x) = (1 – sin(x)^2 + sec(x) * sin(x) * cos(x)) / (cos(x))^3
Now, using the Pythagorean identity sin(x)^2 + cos(x)^2 = 1, we can simplify the numerator:
f'(x) = (1 + sec(x) * sin(x) * cos(x)) / (cos(x))^3
Since sec(x) = 1/cos(x), we can substitute this into the equation:
f'(x) = (1 + sin(x) * cos(x) / cos(x)) / (cos(x))^3
Finally, we can cancel out the cos(x) terms in the numerator:
f'(x) = (1 + sin(x)) / (cos(x))^2
Therefore, the derivative of sec(x) with respect to x is (1 + sin(x)) / (cos(x))^2.
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