Simplifying the Quotient Rule | Derivative of sec(x) with Respect to x = sin(x) / (cos(x))^2

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) with respect to x is given by:

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

In this case, we have f(x) = sec(x), which can be written as f(x) = 1/cos(x). So, g(x) = 1 and h(x) = cos(x).

To find the derivatives g'(x) and h'(x), we differentiate each term separately. Since g(x) = 1, the derivative g'(x) is simply 0.

To find h'(x), we differentiate cos(x) using the chain rule. The derivative of cos(x) is -sin(x), and since we are differentiating with respect to x, we multiply by the derivative of the inside function, which is 1. Thus, h'(x) = -sin(x).

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

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

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

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