Question

Using the quotient rule differentiate the following functions :
1.) sec x
2.) cot x
3.) cosec x

Answer 1

wrong. I'm sure you know the basic derivative formulas. SO why do you keep messing up and not checking your final results?
You correctly got sin(x) / (cos(x))^2
but then said it was sinx secx, rather than secx tanx

come on -- we all know that
d/dx [cot x] = -csc^2(x)

and d/dx [cosec x] = -cscx cotx

Answer 2

1.) For sec x, we know that secant is defined as 1/cos(x). To differentiate this function, we need to use the quotient rule:

d/dx [sec x] = d/dx [1/cos(x)]
= (0*cos(x) - 1*(-sin(x))) / (cos(x))^2
= -(-sin(x)) / (cos(x))^2
= sin(x) / (cos(x))^2
= sin(x)sec(x)

Therefore, the derivative of sec x is sin(x)sec(x).

2.) For cot x, we know that cotangent is defined as cos(x)/sin(x). Using the quotient rule:

d/dx [cot x] = d/dx [cos(x)/sin(x)]
= (sin(x)*(-sin(x)) - cos(x)*cos(x)) / (sin(x))^2
= (-sin^2(x) - cos^2(x)) / (sin^2(x))
= -1 - cos^2(x)/sin^2(x)
= -1 - cos^2(x)csc^2(x)

Therefore, the derivative of cot x is -1 - cos^2(x)csc^2(x).

3.) For cosec x, we know that cosecant is defined as 1/sin(x). Using the quotient rule:

d/dx [cosec x] = d/dx [1/sin(x)]
= (0*sin(x) - 1*cos(x)) / (sin(x))^2
= -cos(x) / (sin(x))^2
= -cos(x)csc(x)

Therefore, the derivative of cosec x is -cos(x)csc(x).

Answer 3

Apologies for the error in my calculations. Here are the correct differentiations using the quotient rule:

1.) Differentiating sec x:
d/dx [sec x] = sin(x)sec(x) = sec(x)tan(x)

2.) Differentiating cot x:
d/dx [cot x] = -csc^2(x)

3.) Differentiating cosec x:
d/dx [cosec x] = -csc(x)cot(x)

Thank you for pointing out the mistake.