The error occurs in step 3. By the product rule, we have
[tex]\dfrac{\mathrm d}{\mathrm dx}(\sec x\times\cos x)=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x+\sec x\times\dfrac{\mathrm d}{\mathrm dx}(\cos x)[/tex]
[tex]=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x\boxed{+\sec x\times(-\sin x)}[/tex]
[tex]=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x\boxed{-\sec x\times\sin x}[/tex]
(i.e. there is a missing factor of [tex]\sin x[/tex])
Then
[tex]\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x-\sec x\times\sin x=0[/tex]
[tex]\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x=\sec x\times\sin x[/tex]
[tex]\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)=\dfrac{\sec x\times\sin x}{\cos x}[/tex]
[tex]\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)=\sec x\times\tan x[/tex]
We want to study the derivation of sec(x) using the product rule.
We will see that the problem is in step 3, so the correct option is C.
We start with step 1, where we see that:
[tex]sec(x)*cos(x) = 1[/tex]
Because this is constant, its derivate will be equal to zero, then we write:
[tex]\frac{d}{dx}(sec(x)*cos(x)) = 0[/tex]
Now we need to apply the product rule, remember that the rule is:
[tex]\frac{d}{dx}(f(x)*g(x)) = ( \frac{df(x)}{dx}*g(x) + \frac{dg(x)}{dx}*f(x))[/tex]
Then we get:
[tex]\frac{d}{dx}(sec(x)*cos(x)) = \frac{d(sec(x))}{dx}*cos(x) + sec(x)*\frac{d(cos(x))}{dx} = 0[/tex]
So here you have your problem, in step 3.
Now we need to remember that the derivate of the cosine function is the negative sine, so we get:
[tex]\frac{d(sec(x))}{dx}*cos(x) - sec(x)*sen(x)= 0\\\\\frac{d(sec(x))}{dx}*cos(x) = sec(x)*sen(x)\\\\\frac{d(sec(x))}{dx} = sec(x)*\frac{sen(x)}{cos(x)} = sec(x)*tan(x)[/tex]
Finally, you got the correct thing, but there is a problem in step 3
If you want to learn more, you can read:
https://brainly.com/question/9964510
