Using the Chain Rule to Find the Derivative of sec(x) with Respect to x

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 if we have a function g(f(x)), then the derivative of g(f(x)) with respect to x is given by g'(f(x)) times f'(x).

We can rewrite sec(x) as 1/cos(x). So, sec(x) = 1/cos(x).

Now, let’s find the derivative of sec(x) using the chain rule:

Let f(x) = cos(x), and g(x) = 1/x.

Applying the chain rule, we have:

(d/dx) sec(x) = (d/dx) (1/cos(x))
= (d/dx) (g(f(x))) [Applying the chain rule]
= g'(f(x)) * f'(x) [Applying the chain rule]

To find g'(f(x)), we differentiate g(x) with respect to x, and substitute f(x) = cos(x):

g'(f(x)) = d/dx (1/f(x))
= d/dx (1/cos(x))
= -sec(x) * tan(x) [Differentiating 1/cos(x) using the quotient rule]

To find f'(x), we differentiate f(x) = cos(x) with respect to x:

f'(x) = d/dx (cos(x))
= -sin(x) [Differentiating cos(x)]

Finally, substituting the values of g'(f(x)) and f'(x) in the chain rule equation, we get:

(d/dx) sec(x) = g'(f(x)) * f'(x)
= (-sec(x) * tan(x)) * (-sin(x))
= sec(x) * tan(x) * sin(x)

So, the derivative of sec(x) with respect to x is sec(x) * tan(x) * sin(x).

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