Calculating the Derivative of sec(x) | The Chain Rule Explained

d/dx(secx)

To find the derivative of sec(x) with respect to x, we can use the chain rule

To find the derivative of sec(x) with respect to x, we can use the chain rule. The chain rule states that for a composite function u(v(x)), the derivative is given by du/dx = du/dv * dv/dx.

Here, we can let u = sec(x) and v = x. Using this notation, we have:

du/dx = du/dv * dv/dx

To find du/dv, we need to differentiate sec(x) with respect to the angle x. The derivative of sec(x) is given by:

du/dv = d/dv(sec(v))

Now, sec(v) can be rewritten as 1/cos(v). Taking the derivative using the quotient rule, we have:

d/dv(1/cos(v)) = (-1/cos^2(v)) * (-sin(v))

Simplifying this further, we get:

du/dv = sin(v)/cos^2(v)

Finally, to obtain the derivative du/dx, we need to multiply du/dv by dv/dx. Since v = x, dv/dx is simply 1. Therefore:

du/dx = du/dv * dv/dx
= sin(v)/cos^2(v) * 1

Substituting v = x back in:

du/dx = sin(x)/cos^2(x)

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

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