How do you derive #y = (x-1)/sin(x)# using the quotient rule?

Answer 1

The derivative is #\frac{sin(x) - (x-1)cos(x)}{sin^2(x)}#

The rule states that

#(f(x)/g(x))' = \frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}#.
So, as long as we know #f(x),g(x),f'(x),g'(x)# and #g^2(x)#, we're ready to write the derivative. Let's compute this quantities:
#f(x)# is of course #x-1#, and #g(x)# is #sin(x)#.
This means that #f'(x)=1#, and #g'(x)=cos(x)#.
Finally, #g^2(x)=sin^2(x)#

Now we have all the "ingredients", and we can put them in the "recipe":

#\frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)} -> \frac{1*sin(x) - (x-1)cos(x)}{sin^2(x)}#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To derive ( y = \frac{x-1}{\sin(x)} ) using the quotient rule, follow these steps:

  1. Identify ( u = x - 1 ) and ( v = \sin(x) ).
  2. Apply the quotient rule formula: ( \frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ).
  3. Differentiate ( u ) and ( v ) with respect to ( x ).
  4. Substitute the derivatives and the original functions into the quotient rule formula.
  5. Simplify the expression if possible.

Following these steps, you'll obtain the derivative of ( y ) with respect to ( x ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To derive (y = \frac{x - 1}{\sin(x)}) using the quotient rule, where (u = x - 1) and (v = \sin(x)), we apply the quotient rule formula:

[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ]

Here, (\frac{du}{dx}) is the derivative of (u) with respect to (x) and (\frac{dv}{dx}) is the derivative of (v) with respect to (x).

First, let's find (\frac{du}{dx}) and (\frac{dv}{dx}):

(\frac{du}{dx} = \frac{d}{dx}(x - 1) = 1)

(\frac{dv}{dx} = \frac{d}{dx}(\sin(x)) = \cos(x))

Now, plug these into the quotient rule formula:

[ \frac{d}{dx}\left(\frac{x - 1}{\sin(x)}\right) = \frac{\sin(x) \cdot 1 - (x - 1) \cdot \cos(x)}{\sin^2(x)} ]

Simplify this expression to get the derivative.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7