How to Find the Derivative of Sec x with Respect to x using Quotient Rule

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 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

Now, let’s see how we can apply this rule to find the derivative of sec x.

We can rewrite sec x as 1 / cos x. So, our function f(x) becomes f(x) = 1 / cos x.

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

Let’s find the derivatives of g(x) and h(x):

g'(x) = 0 (since g(x) = 1, and the derivative of a constant is zero)

h'(x) = -sin x (since the derivative of cos x is -sin x)

Now, we can use the quotient rule 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 x)^2

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

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