d/dx sec x
To find the derivative of sec(x) with respect to x, we need to use the chain rule and the derivative formula for secant
To find the derivative of sec(x) with respect to x, we need to use the chain rule and the derivative formula for secant.
The derivative of sec(x) can be calculated as follows:
Step 1: Recall that the secant function can be expressed as sec(x) = 1/cos(x).
Step 2: Apply the chain rule by considering the derivative of the reciprocal of cos(x).
Let’s denote f(x) = cos(x) and g(x) = 1/f(x).
Step 3: Find the derivative of f(x). The derivative of cos(x) is -sin(x). So, f'(x) = -sin(x).
Step 4: Find the derivative of g(x) using the chain rule. By the chain rule, g'(x) = -f'(x) / [f(x)]^2.
Step 5: Substitute the values we found into the chain rule equation. Therefore, g'(x) = -(-sin(x)) / [cos(x)]^2 = sin(x) / [cos(x)]^2.
Step 6: Finally, substitute back in the original expression for f(x) to find the derivative of sec(x). Recall that f(x) = cos(x). Since g(x) = 1/f(x), we have g(x) = 1/cos(x).
The derivative of sec(x) is therefore sec(x) * tan(x). So, d/dx (sec(x)) = sec(x) * tan(x).
In summary, the derivative of sec(x) is equal to sec(x) times tan(x), derived using the chain rule and the derivative formula for secant.
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