Finding the Derivative of sec(x) with Respect to x: Step-by-Step Guide

(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. 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

In this case, our function is f(x) = sec(x). To apply the quotient rule, we can rewrite sec(x) as 1/cos(x), since sec(x) is the reciprocal of cos(x).

f(x) = 1/cos(x)

Now, let’s identify g(x) and h(x) in our function:

g(x) = 1
h(x) = cos(x)

Next, let’s find the derivatives of g(x) and h(x):

g'(x) = 0 (since 1 is a constant, the derivative is always zero)
h'(x) = -sin(x) (derivative of cos(x))

Now, we can plug these values into the quotient rule formula to find the derivative of f(x):

f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
= [0 * cos(x) – 1 * (-sin(x))] / [cos(x)]^2
= sin(x) / cos^2(x)

Since sin(x) / cos^2(x) can be simplified further, let’s rewrite it using trigonometric identities:

f'(x) = sin(x) / cos^2(x)
= [sin(x) * (1/cos^2(x))] / (1/cos^2(x)) (multiply numerator and denominator by (1/cos^2(x)))
= sin(x) * cos^2(x) / 1
= sin(x) * cos(x)

Therefore, the derivative of sec(x) with respect to x is sin(x) * cos(x).

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