Find the Derivative of sec(x) Using the Quotient Rule

d/dx(secx)

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 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, g(x) = 1 and h(x) = cos(x), since sec(x) is equal to 1/cos(x)

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 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, g(x) = 1 and h(x) = cos(x), since sec(x) is equal to 1/cos(x).

First, let’s find the derivative of g(x):
g'(x) = 0 (since g(x) = 1, a constant)

Next, let’s find the derivative of h(x):
h'(x) = -sin(x) (since the derivative of cos(x) is -sin(x))

Now we can apply the quotient rule:

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(x)]^2

Therefore, the derivative of sec(x) with respect to x is sin(x) / [cos(x)]^2.

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