Derivative of Sec(x): How to Use the Quotient Rule and Simplify the Expression

d/dx secx

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), where g(x) and h(x) are differentiable functions, 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, g(x) = 1 and h(x) = cos(x). So, applying the quotient rule, we have:

sec'(x) = (1*cos(x) – 1*(-sin(x)))/[cos(x)]^2

Simplifying further, we get:

sec'(x) = cos(x) + sin(x)/[cos(x)]^2

To make the expression more elegant, we can rewrite sec(x) using its reciprocal identity:

sec(x) = 1/cos(x)

Then, multiplying sec(x) with the reciprocal of cos(x), we get:

sec(x) * cos(x) = 1

Now, we can substitute this into the equation for sec'(x):

sec'(x) = (cos(x) + sin(x))/[cos(x)]^2

This is the derivative of sec(x) with respect to x, expressed in a simplified form.

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