d/dx(sec(x))
To find the derivative of sec(x), we can use the chain rule
To find the derivative of sec(x), we can use the chain rule. The derivative of sec(x) is mathematically expressed as follows:
d/dx(sec(x)) = sec(x) * tan(x)
Here’s a step-by-step explanation:
1. Recall that sec(x) is the reciprocal of cos(x).
2. Begin by expressing sec(x) as 1/cos(x).
3. Apply the quotient rule, which states that if we have a function f(x) = g(x)/h(x), then f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2. In our case, g(x) = 1 and h(x) = cos(x).
4. Differentiating, g'(x) = 0 (since it is a constant) and h'(x) = -sin(x) using the derivative of cos(x) which is -sin(x).
5. Substitute the values into the quotient rule formula:
[0 * cos(x) – 1 * (-sin(x))] / [cos(x)]^2
= sin(x) / cos^2(x)
6. Remembering that sin(x) / cos(x) is equivalent to tan(x), the final derivative is:
sec(x) * tan(x)
In summary, the derivative of sec(x) is sec(x) * tan(x).
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