Derivative of sec(x)
To find the derivative of sec(x), we can start by expressing sec(x) in terms of sine and cosine functions
To find the derivative of sec(x), we can start by expressing sec(x) in terms of sine and cosine functions.
Recall that sec(x) is the reciprocal of cosine(x), so we can write sec(x) as 1/cos(x).
Next, we can rewrite 1/cos(x) as cos^(-1)(x), where cos^(-1) denotes the reciprocal of cosine.
Now, we can use the chain rule to find the derivative. The chain rule states that if we have a composite function f(g(x)), the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).
Applying the chain rule to cos^(-1)(x), we have f(g(x)) = cos^(-1)(x) and g(x) = cos(x).
The derivative of cos^(-1)(x) with respect to x is -sin(g(x)), where g(x) = cos(x). Therefore, the derivative of cos^(-1)(x) is equal to -sin(cos(x)).
Now, we can find the derivative of sec(x):
d/dx[sec(x)] = d/dx[cos^(-1)(x)]
= d/dx[-sin(cos(x))]
= -sin’cos(x)
= -(cos(x))(-sin(x))
= sin(x)cos(x)
Therefore, the derivative of sec(x) is sin(x)cos(x).
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