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.
Recall that the secant function is defined as: sec x = 1/cos x.
Let’s find the derivative step by step:
1. Start with the function: sec x = 1/cos x.
2. Rewrite sec x as sec x = cos^-1 x.
3. Now we can differentiate using the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2.
Applying the quotient rule, we have:
d/dx [sec x] = d/dx [1/cos x]
Using g(x) = 1 and h(x) = cos x, we find:
g'(x) = 0 (since it is a constant)
h'(x) = -sin x (since the derivative of cos x is -sin x)
4. Substituting these values into the quotient rule formula, we get:
d/dx [sec x] = (0 * cos x – 1 * (-sin x)) / (cos x)^2
Simplifying:
d/dx [sec x] = sin x / (cos x)^2
5. Note that we can also rewrite sin x / (cos x)^2 as sec x * tan x.
So, the final answer is:
d/dx [sec x] = sec x * tan x
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