The Quotient Rule: Find the Derivative of sec(x)

d/dx sec(x)

To find the derivative of sec(x), we can use the quotient rule

To find the derivative of sec(x), we can use the quotient rule.
The quotient rule states that if we have a function f(x) = g(x)/h(x), then its derivative is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

In this case, g(x) = 1 and h(x) = cos(x).

Taking the derivatives of g(x) and h(x) gives us:
g'(x) = 0 (since it is a constant)
h'(x) = -sin(x)

Now we can substitute these values into the quotient rule formula:

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

Next, we can simplify the expression by using the identity sin^2(x) + cos^2(x) = 1:

f'(x) = sin(x) / cos^2(x)
= sin(x) / (1 – sin^2(x))
= sin(x) / cos^2(x)
= tan(x) sec^2(x)

Therefore, the derivative of sec(x) with respect to x is tan(x) sec^2(x).

More Answers: