Understanding the Derivative of the Secant Function | Step-by-Step Quotient Rule Application

d/dx (sec x)

To find the derivative of the secant function with respect to x, we can use the quotient rule

To find the derivative of the secant function with respect to x, we can use the quotient rule. First, recall the quotient rule:

If we have a function f(x) = g(x)/h(x), then the derivative f'(x) is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

Let’s apply the quotient rule to find the derivative of sec(x):

Given f(x) = sec(x), we can rewrite it as f(x) = 1/cos(x) because sec(x) is equal to the reciprocal of cos(x).

Now, let’s identify g(x) and h(x):
g(x) = 1
h(x) = cos(x)

We need to find the derivatives of g(x) and h(x):

g'(x) = 0 (since any constant has a derivative of zero)
h'(x) = -sin(x) (since the derivative of cos(x) is -sin(x))

Now we can substitute the values into the quotient rule:

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

Since sin(x) / cos^2(x) can be further simplified, we can rewrite it as:

f'(x) = tan(x)sec(x)

So, the derivative of sec(x) is tan(x)sec(x).

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